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List of Theorems
RefDescription
dummylink 1 (_Note_: This inference r...
idi 2 Inference form of ~ id . ...
mp2b 9 A double modus ponens infe...
a1i 10 Inference derived from axi...
mp1i 11 Drop and replace an antece...
a2i 12 Inference derived from axi...
imim2i 13 Inference adding common an...
mpd 14 A modus ponens deduction. ...
syl 15 An inference version of th...
mpi 16 A nested modus ponens infe...
mp2 17 A double modus ponens infe...
3syl 18 Inference chaining two syl...
id 19 Principle of identity. Th...
id1 20 Principle of identity. Th...
idd 21 Principle of identity with...
a1d 22 Deduction introducing an e...
a2d 23 Deduction distributing an ...
a1ii 24 Add two antecedents to a w...
sylcom 25 Syllogism inference with c...
syl5com 26 Syllogism inference with c...
com12 27 Inference that swaps (comm...
syl5 28 A syllogism rule of infere...
syl6 29 A syllogism rule of infere...
syl56 30 Combine ~ syl5 and ~ syl6 ...
syl6com 31 Syllogism inference with c...
mpcom 32 Modus ponens inference wit...
syli 33 Syllogism inference with c...
syl2im 34 Replace two antecedents. ...
pm2.27 35 This theorem, called "Asse...
mpdd 36 A nested modus ponens dedu...
mpid 37 A nested modus ponens dedu...
mpdi 38 A nested modus ponens dedu...
mpii 39 A doubly nested modus pone...
syld 40 Syllogism deduction. (Con...
mp2d 41 A double modus ponens dedu...
a1dd 42 Deduction introducing a ne...
pm2.43i 43 Inference absorbing redund...
pm2.43d 44 Deduction absorbing redund...
pm2.43a 45 Inference absorbing redund...
pm2.43b 46 Inference absorbing redund...
pm2.43 47 Absorption of redundant an...
imim2d 48 Deduction adding nested an...
imim2 49 A closed form of syllogism...
embantd 50 Deduction embedding an ant...
3syld 51 Triple syllogism deduction...
sylsyld 52 Virtual deduction rule ~ e...
imim12i 53 Inference joining two impl...
imim1i 54 Inference adding common co...
imim3i 55 Inference adding three nes...
sylc 56 A syllogism inference comb...
syl3c 57 A syllogism inference comb...
syl6mpi 58 ~ e20 without virtual dedu...
mpsyl 59 Modus ponens combined with...
syl6c 60 Inference combining ~ syl6...
syldd 61 Nested syllogism deduction...
syl5d 62 A nested syllogism deducti...
syl7 63 A syllogism rule of infere...
syl6d 64 A nested syllogism deducti...
syl8 65 A syllogism rule of infere...
syl9 66 A nested syllogism inferen...
syl9r 67 A nested syllogism inferen...
imim12d 68 Deduction combining antece...
imim1d 69 Deduction adding nested co...
imim1 70 A closed form of syllogism...
pm2.83 71 Theorem *2.83 of [Whitehea...
com23 72 Commutation of antecedents...
com3r 73 Commutation of antecedents...
com13 74 Commutation of antecedents...
com3l 75 Commutation of antecedents...
pm2.04 76 Swap antecedents. Theorem...
com34 77 Commutation of antecedents...
com4l 78 Commutation of antecedents...
com4t 79 Commutation of antecedents...
com4r 80 Commutation of antecedents...
com24 81 Commutation of antecedents...
com14 82 Commutation of antecedents...
com45 83 Commutation of antecedents...
com35 84 Commutation of antecedents...
com25 85 Commutation of antecedents...
com5l 86 Commutation of antecedents...
com15 87 Commutation of antecedents...
com52l 88 Commutation of antecedents...
com52r 89 Commutation of antecedents...
com5r 90 Commutation of antecedents...
jarr 91 Elimination of a nested an...
pm2.86i 92 Inference based on ~ pm2.8...
pm2.86d 93 Deduction based on ~ pm2.8...
pm2.86 94 Converse of axiom ~ ax-2 ....
loolinALT 95 The Linearity Axiom of the...
loowoz 96 An alternate for the Linea...
con4d 97 Deduction derived from axi...
pm2.21d 98 A contradiction implies an...
pm2.21dd 99 A contradiction implies an...
pm2.21 100 From a wff and its negatio...
pm2.24 101 Theorem *2.24 of [Whitehea...
pm2.18 102 Proof by contradiction. T...
pm2.18d 103 Deduction based on reducti...
notnot2 104 Converse of double negatio...
notnotrd 105 Deduction converting doubl...
notnotri 106 Inference from double nega...
con2d 107 A contraposition deduction...
con2 108 Contraposition. Theorem *...
mt2d 109 Modus tollens deduction. ...
mt2i 110 Modus tollens inference. ...
nsyl3 111 A negated syllogism infere...
con2i 112 A contraposition inference...
nsyl 113 A negated syllogism infere...
notnot1 114 Converse of double negatio...
notnoti 115 Infer double negation. (C...
con1d 116 A contraposition deduction...
mt3d 117 Modus tollens deduction. ...
mt3i 118 Modus tollens inference. ...
nsyl2 119 A negated syllogism infere...
con1 120 Contraposition. Theorem *...
con1i 121 A contraposition inference...
con4i 122 Inference rule derived fro...
pm2.21i 123 A contradiction implies an...
pm2.24ii 124 A contradiction implies an...
con3d 125 A contraposition deduction...
con3 126 Contraposition. Theorem *...
con3i 127 A contraposition inference...
con3rr3 128 Rotate through consequent ...
mt4 129 The rule of modus tollens....
mt4d 130 Modus tollens deduction. ...
mt4i 131 Modus tollens inference. ...
nsyld 132 A negated syllogism deduct...
nsyli 133 A negated syllogism infere...
nsyl4 134 A negated syllogism infere...
pm2.24d 135 Deduction version of ~ pm2...
pm2.24i 136 Inference version of ~ pm2...
pm3.2im 137 Theorem *3.2 of [Whitehead...
mth8 138 Theorem 8 of [Margaris] p....
jc 139 Inference joining the cons...
impi 140 An importation inference. ...
expi 141 An exportation inference. ...
simprim 142 Simplification. Similar t...
simplim 143 Simplification. Similar t...
pm2.5 144 Theorem *2.5 of [Whitehead...
pm2.51 145 Theorem *2.51 of [Whitehea...
pm2.521 146 Theorem *2.521 of [Whitehe...
pm2.52 147 Theorem *2.52 of [Whitehea...
expt 148 Exportation theorem expres...
impt 149 Importation theorem expres...
pm2.61d 150 Deduction eliminating an a...
pm2.61d1 151 Inference eliminating an a...
pm2.61d2 152 Inference eliminating an a...
ja 153 Inference joining the ante...
jad 154 Deduction form of ~ ja . ...
jarl 155 Elimination of a nested an...
pm2.61i 156 Inference eliminating an a...
pm2.61ii 157 Inference eliminating two ...
pm2.61nii 158 Inference eliminating two ...
pm2.61iii 159 Inference eliminating thre...
pm2.01 160 Reductio ad absurdum. The...
pm2.01d 161 Deduction based on reducti...
pm2.6 162 Theorem *2.6 of [Whitehead...
pm2.61 163 Theorem *2.61 of [Whitehea...
pm2.65 164 Theorem *2.65 of [Whitehea...
pm2.65i 165 Inference rule for proof b...
pm2.65d 166 Deduction rule for proof b...
mto 167 The rule of modus tollens....
mtod 168 Modus tollens deduction. ...
mtoi 169 Modus tollens inference. ...
mt2 170 A rule similar to modus to...
mt3 171 A rule similar to modus to...
peirce 172 Peirce's axiom. This odd-...
loolin 173 The Linearity Axiom of the...
looinv 174 The Inversion Axiom of the...
bijust 175 Theorem used to justify de...
bi1 178 Property of the biconditio...
bi3 179 Property of the biconditio...
impbii 180 Infer an equivalence from ...
impbidd 181 Deduce an equivalence from...
impbid21d 182 Deduce an equivalence from...
impbid 183 Deduce an equivalence from...
dfbi1 184 Relate the biconditional c...
dfbi1gb 185 This proof of ~ dfbi1 , di...
biimpi 186 Infer an implication from ...
sylbi 187 A mixed syllogism inferenc...
sylib 188 A mixed syllogism inferenc...
bi2 189 Property of the biconditio...
bicom1 190 Commutative law for equiva...
bicom 191 Commutative law for equiva...
bicomd 192 Commute two sides of a bic...
bicomi 193 Inference from commutative...
impbid1 194 Infer an equivalence from ...
impbid2 195 Infer an equivalence from ...
impcon4bid 196 A variation on ~ impbid wi...
biimpri 197 Infer a converse implicati...
biimpd 198 Deduce an implication from...
mpbi 199 An inference from a bicond...
mpbir 200 An inference from a bicond...
mpbid 201 A deduction from a bicondi...
mpbii 202 An inference from a nested...
sylibr 203 A mixed syllogism inferenc...
sylbir 204 A mixed syllogism inferenc...
sylibd 205 A syllogism deduction. (C...
sylbid 206 A syllogism deduction. (C...
mpbidi 207 A deduction from a bicondi...
syl5bi 208 A mixed syllogism inferenc...
syl5bir 209 A mixed syllogism inferenc...
syl5ib 210 A mixed syllogism inferenc...
syl5ibcom 211 A mixed syllogism inferenc...
syl5ibr 212 A mixed syllogism inferenc...
syl5ibrcom 213 A mixed syllogism inferenc...
biimprd 214 Deduce a converse implicat...
biimpcd 215 Deduce a commuted implicat...
biimprcd 216 Deduce a converse commuted...
syl6ib 217 A mixed syllogism inferenc...
syl6ibr 218 A mixed syllogism inferenc...
syl6bi 219 A mixed syllogism inferenc...
syl6bir 220 A mixed syllogism inferenc...
syl7bi 221 A mixed syllogism inferenc...
syl8ib 222 A syllogism rule of infere...
mpbird 223 A deduction from a bicondi...
mpbiri 224 An inference from a nested...
sylibrd 225 A syllogism deduction. (C...
sylbird 226 A syllogism deduction. (C...
biid 227 Principle of identity for ...
biidd 228 Principle of identity with...
pm5.1im 229 Two propositions are equiv...
2th 230 Two truths are equivalent....
2thd 231 Two truths are equivalent ...
ibi 232 Inference that converts a ...
ibir 233 Inference that converts a ...
ibd 234 Deduction that converts a ...
pm5.74 235 Distribution of implicatio...
pm5.74i 236 Distribution of implicatio...
pm5.74ri 237 Distribution of implicatio...
pm5.74d 238 Distribution of implicatio...
pm5.74rd 239 Distribution of implicatio...
bitri 240 An inference from transiti...
bitr2i 241 An inference from transiti...
bitr3i 242 An inference from transiti...
bitr4i 243 An inference from transiti...
bitrd 244 Deduction form of ~ bitri ...
bitr2d 245 Deduction form of ~ bitr2i...
bitr3d 246 Deduction form of ~ bitr3i...
bitr4d 247 Deduction form of ~ bitr4i...
syl5bb 248 A syllogism inference from...
syl5rbb 249 A syllogism inference from...
syl5bbr 250 A syllogism inference from...
syl5rbbr 251 A syllogism inference from...
syl6bb 252 A syllogism inference from...
syl6rbb 253 A syllogism inference from...
syl6bbr 254 A syllogism inference from...
syl6rbbr 255 A syllogism inference from...
3imtr3i 256 A mixed syllogism inferenc...
3imtr4i 257 A mixed syllogism inferenc...
3imtr3d 258 More general version of ~ ...
3imtr4d 259 More general version of ~ ...
3imtr3g 260 More general version of ~ ...
3imtr4g 261 More general version of ~ ...
3bitri 262 A chained inference from t...
3bitrri 263 A chained inference from t...
3bitr2i 264 A chained inference from t...
3bitr2ri 265 A chained inference from t...
3bitr3i 266 A chained inference from t...
3bitr3ri 267 A chained inference from t...
3bitr4i 268 A chained inference from t...
3bitr4ri 269 A chained inference from t...
3bitrd 270 Deduction from transitivit...
3bitrrd 271 Deduction from transitivit...
3bitr2d 272 Deduction from transitivit...
3bitr2rd 273 Deduction from transitivit...
3bitr3d 274 Deduction from transitivit...
3bitr3rd 275 Deduction from transitivit...
3bitr4d 276 Deduction from transitivit...
3bitr4rd 277 Deduction from transitivit...
3bitr3g 278 More general version of ~ ...
3bitr4g 279 More general version of ~ ...
bi3ant 280 Construct a bi-conditional...
bisym 281 Express symmetries of theo...
notnot 282 Double negation. Theorem ...
con34b 283 Contraposition. Theorem *...
con4bid 284 A contraposition deduction...
notbid 285 Deduction negating both si...
notbi 286 Contraposition. Theorem *...
notbii 287 Negate both sides of a log...
con4bii 288 A contraposition inference...
mtbi 289 An inference from a bicond...
mtbir 290 An inference from a bicond...
mtbid 291 A deduction from a bicondi...
mtbird 292 A deduction from a bicondi...
mtbii 293 An inference from a bicond...
mtbiri 294 An inference from a bicond...
sylnib 295 A mixed syllogism inferenc...
sylnibr 296 A mixed syllogism inferenc...
sylnbi 297 A mixed syllogism inferenc...
sylnbir 298 A mixed syllogism inferenc...
xchnxbi 299 Replacement of a subexpres...
xchnxbir 300 Replacement of a subexpres...
xchbinx 301 Replacement of a subexpres...
xchbinxr 302 Replacement of a subexpres...
imbi2i 303 Introduce an antecedent to...
bibi2i 304 Inference adding a bicondi...
bibi1i 305 Inference adding a bicondi...
bibi12i 306 The equivalence of two equ...
imbi2d 307 Deduction adding an antece...
imbi1d 308 Deduction adding a consequ...
bibi2d 309 Deduction adding a bicondi...
bibi1d 310 Deduction adding a bicondi...
imbi12d 311 Deduction joining two equi...
bibi12d 312 Deduction joining two equi...
imbi1 313 Theorem *4.84 of [Whitehea...
imbi2 314 Theorem *4.85 of [Whitehea...
imbi1i 315 Introduce a consequent to ...
imbi12i 316 Join two logical equivalen...
bibi1 317 Theorem *4.86 of [Whitehea...
con2bi 318 Contraposition. Theorem *...
con2bid 319 A contraposition deduction...
con1bid 320 A contraposition deduction...
con1bii 321 A contraposition inference...
con2bii 322 A contraposition inference...
con1b 323 Contraposition. Bidirecti...
con2b 324 Contraposition. Bidirecti...
biimt 325 A wff is equivalent to its...
pm5.5 326 Theorem *5.5 of [Whitehead...
a1bi 327 Inference rule introducing...
mt2bi 328 A false consequent falsifi...
mtt 329 Modus-tollens-like theorem...
pm5.501 330 Theorem *5.501 of [Whitehe...
ibib 331 Implication in terms of im...
ibibr 332 Implication in terms of im...
tbt 333 A wff is equivalent to its...
nbn2 334 The negation of a wff is e...
bibif 335 Transfer negation via an e...
nbn 336 The negation of a wff is e...
nbn3 337 Transfer falsehood via equ...
pm5.21im 338 Two propositions are equiv...
2false 339 Two falsehoods are equival...
2falsed 340 Two falsehoods are equival...
pm5.21ni 341 Two propositions implying ...
pm5.21nii 342 Eliminate an antecedent im...
pm5.21ndd 343 Eliminate an antecedent im...
bija 344 Combine antecedents into a...
pm5.18 345 Theorem *5.18 of [Whitehea...
xor3 346 Two ways to express "exclu...
nbbn 347 Move negation outside of b...
biass 348 Associative law for the bi...
pm5.19 349 Theorem *5.19 of [Whitehea...
bi2.04 350 Logical equivalence of com...
pm5.4 351 Antecedent absorption impl...
imdi 352 Distributive law for impli...
pm5.41 353 Theorem *5.41 of [Whitehea...
pm4.8 354 Theorem *4.8 of [Whitehead...
pm4.81 355 Theorem *4.81 of [Whitehea...
imim21b 356 Simplify an implication be...
pm4.64 361 Theorem *4.64 of [Whitehea...
pm2.53 362 Theorem *2.53 of [Whitehea...
pm2.54 363 Theorem *2.54 of [Whitehea...
ori 364 Infer implication from dis...
orri 365 Infer implication from dis...
ord 366 Deduce implication from di...
orrd 367 Deduce implication from di...
jaoi 368 Inference disjoining the a...
jaod 369 Deduction disjoining the a...
mpjaod 370 Eliminate a disjunction in...
orel1 371 Elimination of disjunction...
orel2 372 Elimination of disjunction...
olc 373 Introduction of a disjunct...
orc 374 Introduction of a disjunct...
pm1.4 375 Axiom *1.4 of [WhiteheadRu...
orcom 376 Commutative law for disjun...
orcomd 377 Commutation of disjuncts i...
orcoms 378 Commutation of disjuncts i...
orci 379 Deduction introducing a di...
olci 380 Deduction introducing a di...
orcd 381 Deduction introducing a di...
olcd 382 Deduction introducing a di...
orcs 383 Deduction eliminating disj...
olcs 384 Deduction eliminating disj...
pm2.07 385 Theorem *2.07 of [Whitehea...
pm2.45 386 Theorem *2.45 of [Whitehea...
pm2.46 387 Theorem *2.46 of [Whitehea...
pm2.47 388 Theorem *2.47 of [Whitehea...
pm2.48 389 Theorem *2.48 of [Whitehea...
pm2.49 390 Theorem *2.49 of [Whitehea...
pm2.67-2 391 Slight generalization of T...
pm2.67 392 Theorem *2.67 of [Whitehea...
pm2.25 393 Theorem *2.25 of [Whitehea...
biorf 394 A wff is equivalent to its...
biortn 395 A wff is equivalent to its...
biorfi 396 A wff is equivalent to its...
pm2.621 397 Theorem *2.621 of [Whitehe...
pm2.62 398 Theorem *2.62 of [Whitehea...
pm2.68 399 Theorem *2.68 of [Whitehea...
dfor2 400 Logical 'or' expressed in ...
imor 401 Implication in terms of di...
imori 402 Infer disjunction from imp...
imorri 403 Infer implication from dis...
exmid 404 Law of excluded middle, al...
exmidd 405 Law of excluded middle in ...
pm2.1 406 Theorem *2.1 of [Whitehead...
pm2.13 407 Theorem *2.13 of [Whitehea...
pm4.62 408 Theorem *4.62 of [Whitehea...
pm4.66 409 Theorem *4.66 of [Whitehea...
pm4.63 410 Theorem *4.63 of [Whitehea...
imnan 411 Express implication in ter...
imnani 412 Express implication in ter...
iman 413 Express implication in ter...
annim 414 Express conjunction in ter...
pm4.61 415 Theorem *4.61 of [Whitehea...
pm4.65 416 Theorem *4.65 of [Whitehea...
pm4.67 417 Theorem *4.67 of [Whitehea...
imp 418 Importation inference. (C...
impcom 419 Importation inference with...
imp3a 420 Importation deduction. (C...
imp31 421 An importation inference. ...
imp32 422 An importation inference. ...
ex 423 Exportation inference. (T...
expcom 424 Exportation inference with...
exp3a 425 Exportation deduction. (C...
expdimp 426 A deduction version of exp...
impancom 427 Mixed importation/commutat...
con3and 428 Variant of ~ con3d with im...
pm2.01da 429 Deduction based on reducti...
pm2.18da 430 Deduction based on reducti...
pm3.3 431 Theorem *3.3 (Exp) of [Whi...
pm3.31 432 Theorem *3.31 (Imp) of [Wh...
impexp 433 Import-export theorem. Pa...
pm3.2 434 Join antecedents with conj...
pm3.21 435 Join antecedents with conj...
pm3.22 436 Theorem *3.22 of [Whitehea...
ancom 437 Commutative law for conjun...
ancomd 438 Commutation of conjuncts i...
ancoms 439 Inference commuting conjun...
ancomsd 440 Deduction commuting conjun...
pm3.2i 441 Infer conjunction of premi...
pm3.43i 442 Nested conjunction of ante...
simpl 443 Elimination of a conjunct....
simpli 444 Inference eliminating a co...
simpld 445 Deduction eliminating a co...
simplbi 446 Deduction eliminating a co...
simpr 447 Elimination of a conjunct....
simpri 448 Inference eliminating a co...
simprd 449 Deduction eliminating a co...
simprbi 450 Deduction eliminating a co...
adantr 451 Inference adding a conjunc...
adantl 452 Inference adding a conjunc...
adantld 453 Deduction adding a conjunc...
adantrd 454 Deduction adding a conjunc...
mpan9 455 Modus ponens conjoining di...
syldan 456 A syllogism deduction with...
sylan 457 A syllogism inference. (C...
sylanb 458 A syllogism inference. (C...
sylanbr 459 A syllogism inference. (C...
sylan2 460 A syllogism inference. (C...
sylan2b 461 A syllogism inference. (C...
sylan2br 462 A syllogism inference. (C...
syl2an 463 A double syllogism inferen...
syl2anr 464 A double syllogism inferen...
syl2anb 465 A double syllogism inferen...
syl2anbr 466 A double syllogism inferen...
syland 467 A syllogism deduction. (C...
sylan2d 468 A syllogism deduction. (C...
syl2and 469 A syllogism deduction. (C...
biimpa 470 Inference from a logical e...
biimpar 471 Inference from a logical e...
biimpac 472 Inference from a logical e...
biimparc 473 Inference from a logical e...
ianor 474 Negated conjunction in ter...
anor 475 Conjunction in terms of di...
ioran 476 Negated disjunction in ter...
pm4.52 477 Theorem *4.52 of [Whitehea...
pm4.53 478 Theorem *4.53 of [Whitehea...
pm4.54 479 Theorem *4.54 of [Whitehea...
pm4.55 480 Theorem *4.55 of [Whitehea...
pm4.56 481 Theorem *4.56 of [Whitehea...
oran 482 Disjunction in terms of co...
pm4.57 483 Theorem *4.57 of [Whitehea...
pm3.1 484 Theorem *3.1 of [Whitehead...
pm3.11 485 Theorem *3.11 of [Whitehea...
pm3.12 486 Theorem *3.12 of [Whitehea...
pm3.13 487 Theorem *3.13 of [Whitehea...
pm3.14 488 Theorem *3.14 of [Whitehea...
iba 489 Introduction of antecedent...
ibar 490 Introduction of antecedent...
biantru 491 A wff is equivalent to its...
biantrur 492 A wff is equivalent to its...
biantrud 493 A wff is equivalent to its...
biantrurd 494 A wff is equivalent to its...
jaao 495 Inference conjoining and d...
jaoa 496 Inference disjoining and c...
pm3.44 497 Theorem *3.44 of [Whitehea...
jao 498 Disjunction of antecedents...
pm1.2 499 Axiom *1.2 of [WhiteheadRu...
oridm 500 Idempotent law for disjunc...
pm4.25 501 Theorem *4.25 of [Whitehea...
orim12i 502 Disjoin antecedents and co...
orim1i 503 Introduce disjunct to both...
orim2i 504 Introduce disjunct to both...
orbi2i 505 Inference adding a left di...
orbi1i 506 Inference adding a right d...
orbi12i 507 Infer the disjunction of t...
pm1.5 508 Axiom *1.5 (Assoc) of [Whi...
or12 509 Swap two disjuncts. (Cont...
orass 510 Associative law for disjun...
pm2.31 511 Theorem *2.31 of [Whitehea...
pm2.32 512 Theorem *2.32 of [Whitehea...
or32 513 A rearrangement of disjunc...
or4 514 Rearrangement of 4 disjunc...
or42 515 Rearrangement of 4 disjunc...
orordi 516 Distribution of disjunctio...
orordir 517 Distribution of disjunctio...
jca 518 Deduce conjunction of the ...
jcad 519 Deduction conjoining the c...
jca31 520 Join three consequents. (...
jca32 521 Join three consequents. (...
jcai 522 Deduction replacing implic...
jctil 523 Inference conjoining a the...
jctir 524 Inference conjoining a the...
jctl 525 Inference conjoining a the...
jctr 526 Inference conjoining a the...
jctild 527 Deduction conjoining a the...
jctird 528 Deduction conjoining a the...
ancl 529 Conjoin antecedent to left...
anclb 530 Conjoin antecedent to left...
pm5.42 531 Theorem *5.42 of [Whitehea...
ancr 532 Conjoin antecedent to righ...
ancrb 533 Conjoin antecedent to righ...
ancli 534 Deduction conjoining antec...
ancri 535 Deduction conjoining antec...
ancld 536 Deduction conjoining antec...
ancrd 537 Deduction conjoining antec...
anc2l 538 Conjoin antecedent to left...
anc2r 539 Conjoin antecedent to righ...
anc2li 540 Deduction conjoining antec...
anc2ri 541 Deduction conjoining antec...
pm3.41 542 Theorem *3.41 of [Whitehea...
pm3.42 543 Theorem *3.42 of [Whitehea...
pm3.4 544 Conjunction implies implic...
pm4.45im 545 Conjunction with implicati...
anim12d 546 Conjoin antecedents and co...
anim1d 547 Add a conjunct to right of...
anim2d 548 Add a conjunct to left of ...
anim12i 549 Conjoin antecedents and co...
anim12ci 550 Variant of ~ anim12i with ...
anim1i 551 Introduce conjunct to both...
anim2i 552 Introduce conjunct to both...
anim12ii 553 Conjoin antecedents and co...
prth 554 Conjoin antecedents and co...
pm2.3 555 Theorem *2.3 of [Whitehead...
pm2.41 556 Theorem *2.41 of [Whitehea...
pm2.42 557 Theorem *2.42 of [Whitehea...
pm2.4 558 Theorem *2.4 of [Whitehead...
pm2.65da 559 Deduction rule for proof b...
pm4.44 560 Theorem *4.44 of [Whitehea...
pm4.14 561 Theorem *4.14 of [Whitehea...
pm3.37 562 Theorem *3.37 (Transp) of ...
nan 563 Theorem to move a conjunct...
pm4.15 564 Theorem *4.15 of [Whitehea...
pm4.78 565 Theorem *4.78 of [Whitehea...
pm4.79 566 Theorem *4.79 of [Whitehea...
pm4.87 567 Theorem *4.87 of [Whitehea...
pm3.33 568 Theorem *3.33 (Syll) of [W...
pm3.34 569 Theorem *3.34 (Syll) of [W...
pm3.35 570 Conjunctive detachment. T...
pm5.31 571 Theorem *5.31 of [Whitehea...
imp4a 572 An importation inference. ...
imp4b 573 An importation inference. ...
imp4c 574 An importation inference. ...
imp4d 575 An importation inference. ...
imp41 576 An importation inference. ...
imp42 577 An importation inference. ...
imp43 578 An importation inference. ...
imp44 579 An importation inference. ...
imp45 580 An importation inference. ...
imp5a 581 An importation inference. ...
imp5d 582 An importation inference. ...
imp5g 583 An importation inference. ...
imp55 584 An importation inference. ...
imp511 585 An importation inference. ...
expimpd 586 Exportation followed by a ...
exp31 587 An exportation inference. ...
exp32 588 An exportation inference. ...
exp4a 589 An exportation inference. ...
exp4b 590 An exportation inference. ...
exp4c 591 An exportation inference. ...
exp4d 592 An exportation inference. ...
exp41 593 An exportation inference. ...
exp42 594 An exportation inference. ...
exp43 595 An exportation inference. ...
exp44 596 An exportation inference. ...
exp45 597 An exportation inference. ...
expr 598 Export a wff from a right ...
exp5c 599 An exportation inference. ...
exp53 600 An exportation inference. ...
expl 601 Export a wff from a left c...
impr 602 Import a wff into a right ...
impl 603 Export a wff from a left c...
impac 604 Importation with conjuncti...
exbiri 605 Inference form of ~ exbir ...
simprbda 606 Deduction eliminating a co...
simplbda 607 Deduction eliminating a co...
simplbi2 608 Deduction eliminating a co...
dfbi2 609 A theorem similar to the s...
dfbi 610 Definition ~ df-bi rewritt...
pm4.71 611 Implication in terms of bi...
pm4.71r 612 Implication in terms of bi...
pm4.71i 613 Inference converting an im...
pm4.71ri 614 Inference converting an im...
pm4.71d 615 Deduction converting an im...
pm4.71rd 616 Deduction converting an im...
pm5.32 617 Distribution of implicatio...
pm5.32i 618 Distribution of implicatio...
pm5.32ri 619 Distribution of implicatio...
pm5.32d 620 Distribution of implicatio...
pm5.32rd 621 Distribution of implicatio...
pm5.32da 622 Distribution of implicatio...
biadan2 623 Add a conjunction to an eq...
pm4.24 624 Theorem *4.24 of [Whitehea...
anidm 625 Idempotent law for conjunc...
anidms 626 Inference from idempotent ...
anidmdbi 627 Conjunction idempotence wi...
anasss 628 Associative law for conjun...
anassrs 629 Associative law for conjun...
anass 630 Associative law for conjun...
sylanl1 631 A syllogism inference. (C...
sylanl2 632 A syllogism inference. (C...
sylanr1 633 A syllogism inference. (C...
sylanr2 634 A syllogism inference. (C...
sylani 635 A syllogism inference. (C...
sylan2i 636 A syllogism inference. (C...
syl2ani 637 A syllogism inference. (C...
sylan9 638 Nested syllogism inference...
sylan9r 639 Nested syllogism inference...
mtand 640 A modus tollens deduction....
mtord 641 A modus tollens deduction ...
syl2anc 642 Syllogism inference combin...
sylancl 643 Syllogism inference combin...
sylancr 644 Syllogism inference combin...
sylanbrc 645 Syllogism inference. (Con...
sylancb 646 A syllogism inference comb...
sylancbr 647 A syllogism inference comb...
sylancom 648 Syllogism inference with c...
mpdan 649 An inference based on modu...
mpancom 650 An inference based on modu...
mpan 651 An inference based on modu...
mpan2 652 An inference based on modu...
mp2an 653 An inference based on modu...
mp4an 654 An inference based on modu...
mpan2d 655 A deduction based on modus...
mpand 656 A deduction based on modus...
mpani 657 An inference based on modu...
mpan2i 658 An inference based on modu...
mp2ani 659 An inference based on modu...
mp2and 660 A deduction based on modus...
mpanl1 661 An inference based on modu...
mpanl2 662 An inference based on modu...
mpanl12 663 An inference based on modu...
mpanr1 664 An inference based on modu...
mpanr2 665 An inference based on modu...
mpanr12 666 An inference based on modu...
mpanlr1 667 An inference based on modu...
pm5.74da 668 Distribution of implicatio...
pm4.45 669 Theorem *4.45 of [Whitehea...
imdistan 670 Distribution of implicatio...
imdistani 671 Distribution of implicatio...
imdistanri 672 Distribution of implicatio...
imdistand 673 Distribution of implicatio...
imdistanda 674 Distribution of implicatio...
anbi2i 675 Introduce a left conjunct ...
anbi1i 676 Introduce a right conjunct...
anbi2ci 677 Variant of ~ anbi2i with c...
anbi12i 678 Conjoin both sides of two ...
anbi12ci 679 Variant of ~ anbi12i with ...
sylan9bb 680 Nested syllogism inference...
sylan9bbr 681 Nested syllogism inference...
orbi2d 682 Deduction adding a left di...
orbi1d 683 Deduction adding a right d...
anbi2d 684 Deduction adding a left co...
anbi1d 685 Deduction adding a right c...
orbi1 686 Theorem *4.37 of [Whitehea...
anbi1 687 Introduce a right conjunct...
anbi2 688 Introduce a left conjunct ...
bitr 689 Theorem *4.22 of [Whitehea...
orbi12d 690 Deduction joining two equi...
anbi12d 691 Deduction joining two equi...
pm5.3 692 Theorem *5.3 of [Whitehead...
pm5.61 693 Theorem *5.61 of [Whitehea...
adantll 694 Deduction adding a conjunc...
adantlr 695 Deduction adding a conjunc...
adantrl 696 Deduction adding a conjunc...
adantrr 697 Deduction adding a conjunc...
adantlll 698 Deduction adding a conjunc...
adantllr 699 Deduction adding a conjunc...
adantlrl 700 Deduction adding a conjunc...
adantlrr 701 Deduction adding a conjunc...
adantrll 702 Deduction adding a conjunc...
adantrlr 703 Deduction adding a conjunc...
adantrrl 704 Deduction adding a conjunc...
adantrrr 705 Deduction adding a conjunc...
ad2antrr 706 Deduction adding two conju...
ad2antlr 707 Deduction adding two conju...
ad2antrl 708 Deduction adding two conju...
ad2antll 709 Deduction adding conjuncts...
ad3antrrr 710 Deduction adding three con...
ad3antlr 711 Deduction adding three con...
ad4antr 712 Deduction adding 4 conjunc...
ad4antlr 713 Deduction adding 4 conjunc...
ad5antr 714 Deduction adding 5 conjunc...
ad5antlr 715 Deduction adding 5 conjunc...
ad6antr 716 Deduction adding 6 conjunc...
ad6antlr 717 Deduction adding 6 conjunc...
ad7antr 718 Deduction adding 7 conjunc...
ad7antlr 719 Deduction adding 7 conjunc...
ad8antr 720 Deduction adding 8 conjunc...
ad8antlr 721 Deduction adding 8 conjunc...
ad9antr 722 Deduction adding 9 conjunc...
ad9antlr 723 Deduction adding 9 conjunc...
ad10antr 724 Deduction adding 10 conjun...
ad10antlr 725 Deduction adding 10 conjun...
ad2ant2l 726 Deduction adding two conju...
ad2ant2r 727 Deduction adding two conju...
ad2ant2lr 728 Deduction adding two conju...
ad2ant2rl 729 Deduction adding two conju...
simpll 730 Simplification of a conjun...
simplr 731 Simplification of a conjun...
simprl 732 Simplification of a conjun...
simprr 733 Simplification of a conjun...
simplll 734 Simplification of a conjun...
simpllr 735 Simplification of a conjun...
simplrl 736 Simplification of a conjun...
simplrr 737 Simplification of a conjun...
simprll 738 Simplification of a conjun...
simprlr 739 Simplification of a conjun...
simprrl 740 Simplification of a conjun...
simprrr 741 Simplification of a conjun...
simp-4l 742 Simplification of a conjun...
simp-4r 743 Simplification of a conjun...
simp-5l 744 Simplification of a conjun...
simp-5r 745 Simplification of a conjun...
simp-6l 746 Simplification of a conjun...
simp-6r 747 Simplification of a conjun...
simp-7l 748 Simplification of a conjun...
simp-7r 749 Simplification of a conjun...
simp-8l 750 Simplification of a conjun...
simp-8r 751 Simplification of a conjun...
simp-9l 752 Simplification of a conjun...
simp-9r 753 Simplification of a conjun...
simp-10l 754 Simplification of a conjun...
simp-10r 755 Simplification of a conjun...
simp-11l 756 Simplification of a conjun...
simp-11r 757 Simplification of a conjun...
jaob 758 Disjunction of antecedents...
jaoian 759 Inference disjoining the a...
jaodan 760 Deduction disjoining the a...
mpjaodan 761 Eliminate a disjunction in...
pm4.77 762 Theorem *4.77 of [Whitehea...
pm2.63 763 Theorem *2.63 of [Whitehea...
pm2.64 764 Theorem *2.64 of [Whitehea...
pm2.61ian 765 Elimination of an antecede...
pm2.61dan 766 Elimination of an antecede...
pm2.61ddan 767 Elimination of two anteced...
pm2.61dda 768 Elimination of two anteced...
condan 769 Proof by contradiction. (...
abai 770 Introduce one conjunct as ...
pm5.53 771 Theorem *5.53 of [Whitehea...
an12 772 Swap two conjuncts. Note ...
an32 773 A rearrangement of conjunc...
an13 774 A rearrangement of conjunc...
an31 775 A rearrangement of conjunc...
an12s 776 Swap two conjuncts in ante...
ancom2s 777 Inference commuting a nest...
an13s 778 Swap two conjuncts in ante...
an32s 779 Swap two conjuncts in ante...
ancom1s 780 Inference commuting a nest...
an31s 781 Swap two conjuncts in ante...
anass1rs 782 Commutative-associative la...
anabs1 783 Absorption into embedded c...
anabs5 784 Absorption into embedded c...
anabs7 785 Absorption into embedded c...
anabsan 786 Absorption of antecedent w...
anabss1 787 Absorption of antecedent i...
anabss4 788 Absorption of antecedent i...
anabss5 789 Absorption of antecedent i...
anabsi5 790 Absorption of antecedent i...
anabsi6 791 Absorption of antecedent i...
anabsi7 792 Absorption of antecedent i...
anabsi8 793 Absorption of antecedent i...
anabss7 794 Absorption of antecedent i...
anabsan2 795 Absorption of antecedent w...
anabss3 796 Absorption of antecedent i...
an4 797 Rearrangement of 4 conjunc...
an42 798 Rearrangement of 4 conjunc...
an4s 799 Inference rearranging 4 co...
an42s 800 Inference rearranging 4 co...
anandi 801 Distribution of conjunctio...
anandir 802 Distribution of conjunctio...
anandis 803 Inference that undistribut...
anandirs 804 Inference that undistribut...
impbida 805 Deduce an equivalence from...
pm3.48 806 Theorem *3.48 of [Whitehea...
pm3.45 807 Theorem *3.45 (Fact) of [W...
im2anan9 808 Deduction joining nested i...
im2anan9r 809 Deduction joining nested i...
anim12dan 810 Conjoin antecedents and co...
orim12d 811 Disjoin antecedents and co...
orim1d 812 Disjoin antecedents and co...
orim2d 813 Disjoin antecedents and co...
orim2 814 Axiom *1.6 (Sum) of [White...
pm2.38 815 Theorem *2.38 of [Whitehea...
pm2.36 816 Theorem *2.36 of [Whitehea...
pm2.37 817 Theorem *2.37 of [Whitehea...
pm2.73 818 Theorem *2.73 of [Whitehea...
pm2.74 819 Theorem *2.74 of [Whitehea...
orimdi 820 Disjunction distributes ov...
pm2.76 821 Theorem *2.76 of [Whitehea...
pm2.75 822 Theorem *2.75 of [Whitehea...
pm2.8 823 Theorem *2.8 of [Whitehead...
pm2.81 824 Theorem *2.81 of [Whitehea...
pm2.82 825 Theorem *2.82 of [Whitehea...
pm2.85 826 Theorem *2.85 of [Whitehea...
pm3.2ni 827 Infer negated disjunction ...
orabs 828 Absorption of redundant in...
oranabs 829 Absorb a disjunct into a c...
pm5.1 830 Two propositions are equiv...
pm5.21 831 Two propositions are equiv...
pm3.43 832 Theorem *3.43 (Comp) of [W...
jcab 833 Distributive law for impli...
ordi 834 Distributive law for disju...
ordir 835 Distributive law for disju...
pm4.76 836 Theorem *4.76 of [Whitehea...
andi 837 Distributive law for conju...
andir 838 Distributive law for conju...
orddi 839 Double distributive law fo...
anddi 840 Double distributive law fo...
pm4.39 841 Theorem *4.39 of [Whitehea...
pm4.38 842 Theorem *4.38 of [Whitehea...
bi2anan9 843 Deduction joining two equi...
bi2anan9r 844 Deduction joining two equi...
bi2bian9 845 Deduction joining two bico...
pm4.72 846 Implication in terms of bi...
imimorb 847 Simplify an implication be...
pm5.33 848 Theorem *5.33 of [Whitehea...
pm5.36 849 Theorem *5.36 of [Whitehea...
bianabs 850 Absorb a hypothesis into t...
oibabs 851 Absorption of disjunction ...
pm3.24 852 Law of noncontradiction. ...
pm2.26 853 Theorem *2.26 of [Whitehea...
pm5.11 854 Theorem *5.11 of [Whitehea...
pm5.12 855 Theorem *5.12 of [Whitehea...
pm5.14 856 Theorem *5.14 of [Whitehea...
pm5.13 857 Theorem *5.13 of [Whitehea...
pm5.17 858 Theorem *5.17 of [Whitehea...
pm5.15 859 Theorem *5.15 of [Whitehea...
pm5.16 860 Theorem *5.16 of [Whitehea...
xor 861 Two ways to express "exclu...
nbi2 862 Two ways to express "exclu...
dfbi3 863 An alternate definition of...
pm5.24 864 Theorem *5.24 of [Whitehea...
xordi 865 Conjunction distributes ov...
biort 866 A wff disjoined with truth...
pm5.55 867 Theorem *5.55 of [Whitehea...
pm5.21nd 868 Eliminate an antecedent im...
pm5.35 869 Theorem *5.35 of [Whitehea...
pm5.54 870 Theorem *5.54 of [Whitehea...
baib 871 Move conjunction outside o...
baibr 872 Move conjunction outside o...
rbaib 873 Move conjunction outside o...
rbaibr 874 Move conjunction outside o...
baibd 875 Move conjunction outside o...
rbaibd 876 Move conjunction outside o...
pm5.44 877 Theorem *5.44 of [Whitehea...
pm5.6 878 Conjunction in antecedent ...
orcanai 879 Change disjunction in cons...
intnan 880 Introduction of conjunct i...
intnanr 881 Introduction of conjunct i...
intnand 882 Introduction of conjunct i...
intnanrd 883 Introduction of conjunct i...
mpbiran 884 Detach truth from conjunct...
mpbiran2 885 Detach truth from conjunct...
mpbir2an 886 Detach a conjunction of tr...
mpbi2and 887 Detach a conjunction of tr...
mpbir2and 888 Detach a conjunction of tr...
pm5.62 889 Theorem *5.62 of [Whitehea...
pm5.63 890 Theorem *5.63 of [Whitehea...
bianfi 891 A wff conjoined with false...
bianfd 892 A wff conjoined with false...
pm4.43 893 Theorem *4.43 of [Whitehea...
pm4.82 894 Theorem *4.82 of [Whitehea...
pm4.83 895 Theorem *4.83 of [Whitehea...
pclem6 896 Negation inferred from emb...
biantr 897 A transitive law of equiva...
orbidi 898 Disjunction distributes ov...
biluk 899 Lukasiewicz's shortest axi...
pm5.7 900 Disjunction distributes ov...
bigolden 901 Dijkstra-Scholten's Golden...
pm5.71 902 Theorem *5.71 of [Whitehea...
pm5.75 903 Theorem *5.75 of [Whitehea...
bimsc1 904 Removal of conjunct from o...
4exmid 905 The disjunction of the fou...
ecase2d 906 Deduction for elimination ...
ecase3 907 Inference for elimination ...
ecase 908 Inference for elimination ...
ecase3d 909 Deduction for elimination ...
ecased 910 Deduction for elimination ...
ecase3ad 911 Deduction for elimination ...
ccase 912 Inference for combining ca...
ccased 913 Deduction for combining ca...
ccase2 914 Inference for combining ca...
4cases 915 Inference eliminating two ...
4casesdan 916 Deduction eliminating two ...
niabn 917 Miscellaneous inference re...
dedlem0a 918 Lemma for an alternate ver...
dedlem0b 919 Lemma for an alternate ver...
dedlema 920 Lemma for weak deduction t...
dedlemb 921 Lemma for weak deduction t...
elimh 922 Hypothesis builder for wea...
dedt 923 The weak deduction theorem...
con3th 924 Contraposition. Theorem *...
consensus 925 The consensus theorem. Th...
pm4.42 926 Theorem *4.42 of [Whitehea...
ninba 927 Miscellaneous inference re...
prlem1 928 A specialized lemma for se...
prlem2 929 A specialized lemma for se...
oplem1 930 A specialized lemma for se...
rnlem 931 Lemma used in construction...
dn1 932 A single axiom for Boolean...
3orass 937 Associative law for triple...
3anass 938 Associative law for triple...
3anrot 939 Rotation law for triple co...
3orrot 940 Rotation law for triple di...
3ancoma 941 Commutation law for triple...
3orcoma 942 Commutation law for triple...
3ancomb 943 Commutation law for triple...
3orcomb 944 Commutation law for triple...
3anrev 945 Reversal law for triple co...
3anan32 946 Convert triple conjunction...
3anan12 947 Convert triple conjunction...
3anor 948 Triple conjunction express...
3ianor 949 Negated triple conjunction...
3ioran 950 Negated triple disjunction...
3oran 951 Triple disjunction in term...
3simpa 952 Simplification of triple c...
3simpb 953 Simplification of triple c...
3simpc 954 Simplification of triple c...
simp1 955 Simplification of triple c...
simp2 956 Simplification of triple c...
simp3 957 Simplification of triple c...
simpl1 958 Simplification rule. (Con...
simpl2 959 Simplification rule. (Con...
simpl3 960 Simplification rule. (Con...
simpr1 961 Simplification rule. (Con...
simpr2 962 Simplification rule. (Con...
simpr3 963 Simplification rule. (Con...
simp1i 964 Infer a conjunct from a tr...
simp2i 965 Infer a conjunct from a tr...
simp3i 966 Infer a conjunct from a tr...
simp1d 967 Deduce a conjunct from a t...
simp2d 968 Deduce a conjunct from a t...
simp3d 969 Deduce a conjunct from a t...
simp1bi 970 Deduce a conjunct from a t...
simp2bi 971 Deduce a conjunct from a t...
simp3bi 972 Deduce a conjunct from a t...
3adant1 973 Deduction adding a conjunc...
3adant2 974 Deduction adding a conjunc...
3adant3 975 Deduction adding a conjunc...
3ad2ant1 976 Deduction adding conjuncts...
3ad2ant2 977 Deduction adding conjuncts...
3ad2ant3 978 Deduction adding conjuncts...
simp1l 979 Simplification of triple c...
simp1r 980 Simplification of triple c...
simp2l 981 Simplification of triple c...
simp2r 982 Simplification of triple c...
simp3l 983 Simplification of triple c...
simp3r 984 Simplification of triple c...
simp11 985 Simplification of doubly t...
simp12 986 Simplification of doubly t...
simp13 987 Simplification of doubly t...
simp21 988 Simplification of doubly t...
simp22 989 Simplification of doubly t...
simp23 990 Simplification of doubly t...
simp31 991 Simplification of doubly t...
simp32 992 Simplification of doubly t...
simp33 993 Simplification of doubly t...
simpll1 994 Simplification of conjunct...
simpll2 995 Simplification of conjunct...
simpll3 996 Simplification of conjunct...
simplr1 997 Simplification of conjunct...
simplr2 998 Simplification of conjunct...
simplr3 999 Simplification of conjunct...
simprl1 1000 Simplification of conjunct...
simprl2 1001 Simplification of conjunct...
simprl3 1002 Simplification of conjunct...
simprr1 1003 Simplification of conjunct...
simprr2 1004 Simplification of conjunct...
simprr3 1005 Simplification of conjunct...
simpl1l 1006 Simplification of conjunct...
simpl1r 1007 Simplification of conjunct...
simpl2l 1008 Simplification of conjunct...
simpl2r 1009 Simplification of conjunct...
simpl3l 1010 Simplification of conjunct...
simpl3r 1011 Simplification of conjunct...
simpr1l 1012 Simplification of conjunct...
simpr1r 1013 Simplification of conjunct...
simpr2l 1014 Simplification of conjunct...
simpr2r 1015 Simplification of conjunct...
simpr3l 1016 Simplification of conjunct...
simpr3r 1017 Simplification of conjunct...
simp1ll 1018 Simplification of conjunct...
simp1lr 1019 Simplification of conjunct...
simp1rl 1020 Simplification of conjunct...
simp1rr 1021 Simplification of conjunct...
simp2ll 1022 Simplification of conjunct...
simp2lr 1023 Simplification of conjunct...
simp2rl 1024 Simplification of conjunct...
simp2rr 1025 Simplification of conjunct...
simp3ll 1026 Simplification of conjunct...
simp3lr 1027 Simplification of conjunct...
simp3rl 1028 Simplification of conjunct...
simp3rr 1029 Simplification of conjunct...
simpl11 1030 Simplification of conjunct...
simpl12 1031 Simplification of conjunct...
simpl13 1032 Simplification of conjunct...
simpl21 1033 Simplification of conjunct...
simpl22 1034 Simplification of conjunct...
simpl23 1035 Simplification of conjunct...
simpl31 1036 Simplification of conjunct...
simpl32 1037 Simplification of conjunct...
simpl33 1038 Simplification of conjunct...
simpr11 1039 Simplification of conjunct...
simpr12 1040 Simplification of conjunct...
simpr13 1041 Simplification of conjunct...
simpr21 1042 Simplification of conjunct...
simpr22 1043 Simplification of conjunct...
simpr23 1044 Simplification of conjunct...
simpr31 1045 Simplification of conjunct...
simpr32 1046 Simplification of conjunct...
simpr33 1047 Simplification of conjunct...
simp1l1 1048 Simplification of conjunct...
simp1l2 1049 Simplification of conjunct...
simp1l3 1050 Simplification of conjunct...
simp1r1 1051 Simplification of conjunct...
simp1r2 1052 Simplification of conjunct...
simp1r3 1053 Simplification of conjunct...
simp2l1 1054 Simplification of conjunct...
simp2l2 1055 Simplification of conjunct...
simp2l3 1056 Simplification of conjunct...
simp2r1 1057 Simplification of conjunct...
simp2r2 1058 Simplification of conjunct...
simp2r3 1059 Simplification of conjunct...
simp3l1 1060 Simplification of conjunct...
simp3l2 1061 Simplification of conjunct...
simp3l3 1062 Simplification of conjunct...
simp3r1 1063 Simplification of conjunct...
simp3r2 1064 Simplification of conjunct...
simp3r3 1065 Simplification of conjunct...
simp11l 1066 Simplification of conjunct...
simp11r 1067 Simplification of conjunct...
simp12l 1068 Simplification of conjunct...
simp12r 1069 Simplification of conjunct...
simp13l 1070 Simplification of conjunct...
simp13r 1071 Simplification of conjunct...
simp21l 1072 Simplification of conjunct...
simp21r 1073 Simplification of conjunct...
simp22l 1074 Simplification of conjunct...
simp22r 1075 Simplification of conjunct...
simp23l 1076 Simplification of conjunct...
simp23r 1077 Simplification of conjunct...
simp31l 1078 Simplification of conjunct...
simp31r 1079 Simplification of conjunct...
simp32l 1080 Simplification of conjunct...
simp32r 1081 Simplification of conjunct...
simp33l 1082 Simplification of conjunct...
simp33r 1083 Simplification of conjunct...
simp111 1084 Simplification of conjunct...
simp112 1085 Simplification of conjunct...
simp113 1086 Simplification of conjunct...
simp121 1087 Simplification of conjunct...
simp122 1088 Simplification of conjunct...
simp123 1089 Simplification of conjunct...
simp131 1090 Simplification of conjunct...
simp132 1091 Simplification of conjunct...
simp133 1092 Simplification of conjunct...
simp211 1093 Simplification of conjunct...
simp212 1094 Simplification of conjunct...
simp213 1095 Simplification of conjunct...
simp221 1096 Simplification of conjunct...
simp222 1097 Simplification of conjunct...
simp223 1098 Simplification of conjunct...
simp231 1099 Simplification of conjunct...
simp232 1100 Simplification of conjunct...
simp233 1101 Simplification of conjunct...
simp311 1102 Simplification of conjunct...
simp312 1103 Simplification of conjunct...
simp313 1104 Simplification of conjunct...
simp321 1105 Simplification of conjunct...
simp322 1106 Simplification of conjunct...
simp323 1107 Simplification of conjunct...
simp331 1108 Simplification of conjunct...
simp332 1109 Simplification of conjunct...
simp333 1110 Simplification of conjunct...
3adantl1 1111 Deduction adding a conjunc...
3adantl2 1112 Deduction adding a conjunc...
3adantl3 1113 Deduction adding a conjunc...
3adantr1 1114 Deduction adding a conjunc...
3adantr2 1115 Deduction adding a conjunc...
3adantr3 1116 Deduction adding a conjunc...
3ad2antl1 1117 Deduction adding conjuncts...
3ad2antl2 1118 Deduction adding conjuncts...
3ad2antl3 1119 Deduction adding conjuncts...
3ad2antr1 1120 Deduction adding conjuncts...
3ad2antr2 1121 Deduction adding conjuncts...
3ad2antr3 1122 Deduction adding conjuncts...
3anibar 1123 Remove a hypothesis from t...
3mix1 1124 Introduction in triple dis...
3mix2 1125 Introduction in triple dis...
3mix3 1126 Introduction in triple dis...
3mix1i 1127 Introduction in triple dis...
3mix2i 1128 Introduction in triple dis...
3mix3i 1129 Introduction in triple dis...
3pm3.2i 1130 Infer conjunction of premi...
pm3.2an3 1131 ~ pm3.2 for a triple conju...
3jca 1132 Join consequents with conj...
3jcad 1133 Deduction conjoining the c...
mpbir3an 1134 Detach a conjunction of tr...
mpbir3and 1135 Detach a conjunction of tr...
syl3anbrc 1136 Syllogism inference. (Con...
3anim123i 1137 Join antecedents and conse...
3anim1i 1138 Add two conjuncts to antec...
3anim3i 1139 Add two conjuncts to antec...
3anbi123i 1140 Join 3 biconditionals with...
3orbi123i 1141 Join 3 biconditionals with...
3anbi1i 1142 Inference adding two conju...
3anbi2i 1143 Inference adding two conju...
3anbi3i 1144 Inference adding two conju...
3imp 1145 Importation inference. (C...
3impa 1146 Importation from double to...
3impb 1147 Importation from double to...
3impia 1148 Importation to triple conj...
3impib 1149 Importation to triple conj...
3exp 1150 Exportation inference. (C...
3expa 1151 Exportation from triple to...
3expb 1152 Exportation from triple to...
3expia 1153 Exportation from triple co...
3expib 1154 Exportation from triple co...
3com12 1155 Commutation in antecedent....
3com13 1156 Commutation in antecedent....
3com23 1157 Commutation in antecedent....
3coml 1158 Commutation in antecedent....
3comr 1159 Commutation in antecedent....
3adant3r1 1160 Deduction adding a conjunc...
3adant3r2 1161 Deduction adding a conjunc...
3adant3r3 1162 Deduction adding a conjunc...
3an1rs 1163 Swap conjuncts. (Contribu...
3imp1 1164 Importation to left triple...
3impd 1165 Importation deduction for ...
3imp2 1166 Importation to right tripl...
3exp1 1167 Exportation from left trip...
3expd 1168 Exportation deduction for ...
3exp2 1169 Exportation from right tri...
exp5o 1170 A triple exportation infer...
exp516 1171 A triple exportation infer...
exp520 1172 A triple exportation infer...
3anassrs 1173 Associative law for conjun...
3adant1l 1174 Deduction adding a conjunc...
3adant1r 1175 Deduction adding a conjunc...
3adant2l 1176 Deduction adding a conjunc...
3adant2r 1177 Deduction adding a conjunc...
3adant3l 1178 Deduction adding a conjunc...
3adant3r 1179 Deduction adding a conjunc...
syl12anc 1180 Syllogism combined with co...
syl21anc 1181 Syllogism combined with co...
syl3anc 1182 Syllogism combined with co...
syl22anc 1183 Syllogism combined with co...
syl13anc 1184 Syllogism combined with co...
syl31anc 1185 Syllogism combined with co...
syl112anc 1186 Syllogism combined with co...
syl121anc 1187 Syllogism combined with co...
syl211anc 1188 Syllogism combined with co...
syl23anc 1189 Syllogism combined with co...
syl32anc 1190 Syllogism combined with co...
syl122anc 1191 Syllogism combined with co...
syl212anc 1192 Syllogism combined with co...
syl221anc 1193 Syllogism combined with co...
syl113anc 1194 Syllogism combined with co...
syl131anc 1195 Syllogism combined with co...
syl311anc 1196 Syllogism combined with co...
syl33anc 1197 Syllogism combined with co...
syl222anc 1198 Syllogism combined with co...
syl123anc 1199 Syllogism combined with co...
syl132anc 1200 Syllogism combined with co...
syl213anc 1201 Syllogism combined with co...
syl231anc 1202 Syllogism combined with co...
syl312anc 1203 Syllogism combined with co...
syl321anc 1204 Syllogism combined with co...
syl133anc 1205 Syllogism combined with co...
syl313anc 1206 Syllogism combined with co...
syl331anc 1207 Syllogism combined with co...
syl223anc 1208 Syllogism combined with co...
syl232anc 1209 Syllogism combined with co...
syl322anc 1210 Syllogism combined with co...
syl233anc 1211 Syllogism combined with co...
syl323anc 1212 Syllogism combined with co...
syl332anc 1213 Syllogism combined with co...
syl333anc 1214 A syllogism inference comb...
syl3an1 1215 A syllogism inference. (C...
syl3an2 1216 A syllogism inference. (C...
syl3an3 1217 A syllogism inference. (C...
syl3an1b 1218 A syllogism inference. (C...
syl3an2b 1219 A syllogism inference. (C...
syl3an3b 1220 A syllogism inference. (C...
syl3an1br 1221 A syllogism inference. (C...
syl3an2br 1222 A syllogism inference. (C...
syl3an3br 1223 A syllogism inference. (C...
syl3an 1224 A triple syllogism inferen...
syl3anb 1225 A triple syllogism inferen...
syl3anbr 1226 A triple syllogism inferen...
syld3an3 1227 A syllogism inference. (C...
syld3an1 1228 A syllogism inference. (C...
syld3an2 1229 A syllogism inference. (C...
syl3anl1 1230 A syllogism inference. (C...
syl3anl2 1231 A syllogism inference. (C...
syl3anl3 1232 A syllogism inference. (C...
syl3anl 1233 A triple syllogism inferen...
syl3anr1 1234 A syllogism inference. (C...
syl3anr2 1235 A syllogism inference. (C...
syl3anr3 1236 A syllogism inference. (C...
3impdi 1237 Importation inference (und...
3impdir 1238 Importation inference (und...
3anidm12 1239 Inference from idempotent ...
3anidm13 1240 Inference from idempotent ...
3anidm23 1241 Inference from idempotent ...
3ori 1242 Infer implication from tri...
3jao 1243 Disjunction of 3 anteceden...
3jaob 1244 Disjunction of 3 anteceden...
3jaoi 1245 Disjunction of 3 anteceden...
3jaod 1246 Disjunction of 3 anteceden...
3jaoian 1247 Disjunction of 3 anteceden...
3jaodan 1248 Disjunction of 3 anteceden...
3jaao 1249 Inference conjoining and d...
syl3an9b 1250 Nested syllogism inference...
3orbi123d 1251 Deduction joining 3 equiva...
3anbi123d 1252 Deduction joining 3 equiva...
3anbi12d 1253 Deduction conjoining and a...
3anbi13d 1254 Deduction conjoining and a...
3anbi23d 1255 Deduction conjoining and a...
3anbi1d 1256 Deduction adding conjuncts...
3anbi2d 1257 Deduction adding conjuncts...
3anbi3d 1258 Deduction adding conjuncts...
3anim123d 1259 Deduction joining 3 implic...
3orim123d 1260 Deduction joining 3 implic...
an6 1261 Rearrangement of 6 conjunc...
3an6 1262 Analog of ~ an4 for triple...
3or6 1263 Analog of ~ or4 for triple...
mp3an1 1264 An inference based on modu...
mp3an2 1265 An inference based on modu...
mp3an3 1266 An inference based on modu...
mp3an12 1267 An inference based on modu...
mp3an13 1268 An inference based on modu...
mp3an23 1269 An inference based on modu...
mp3an1i 1270 An inference based on modu...
mp3anl1 1271 An inference based on modu...
mp3anl2 1272 An inference based on modu...
mp3anl3 1273 An inference based on modu...
mp3anr1 1274 An inference based on modu...
mp3anr2 1275 An inference based on modu...
mp3anr3 1276 An inference based on modu...
mp3an 1277 An inference based on modu...
mpd3an3 1278 An inference based on modu...
mpd3an23 1279 An inference based on modu...
mp3and 1280 A deduction based on modus...
biimp3a 1281 Infer implication from a l...
biimp3ar 1282 Infer implication from a l...
3anandis 1283 Inference that undistribut...
3anandirs 1284 Inference that undistribut...
ecase23d 1285 Deduction for elimination ...
3ecase 1286 Inference for elimination ...
nanan 1289 Write 'and' in terms of 'n...
nancom 1290 The 'nand' operator commut...
nannan 1291 Lemma for handling nested ...
nanim 1292 Show equivalence between i...
nannot 1293 Show equivalence between n...
nanbi 1294 Show equivalence between t...
xnor 1297 Two ways to write XNOR. (C...
xorcom 1298 ` \/_ ` is commutative. (...
xorass 1299 ` \/_ ` is associative. (...
excxor 1300 This tautology shows that ...
xor2 1301 Two ways to express "exclu...
xorneg1 1302 ` \/_ ` is negated under n...
xorneg2 1303 ` \/_ ` is negated under n...
xorneg 1304 ` \/_ ` is unchanged under...
xorbi12i 1305 Equality property for XOR....
xorbi12d 1306 Equality property for XOR....
trujust 1309 Soundness justification th...
tru 1312 ` T. ` is provable. (Cont...
fal 1313 ` F. ` is refutable. (Con...
trud 1314 Eliminate ` T. ` as an ant...
tbtru 1315 If something is true, it o...
nbfal 1316 If something is not true, ...
bitru 1317 A theorem is equivalent to...
bifal 1318 A contradiction is equival...
falim 1319 ` F. ` implies anything. ...
falimd 1320 ` F. ` implies anything. ...
a1tru 1321 Anything implies ` T. ` . ...
dfnot 1322 Given falsum, we can defin...
inegd 1323 Negation introduction rule...
efald 1324 Deduction based on reducti...
pm2.21fal 1325 If a wff and its negation ...
truantru 1326 A ` /\ ` identity. (Contr...
truanfal 1327 A ` /\ ` identity. (Contr...
falantru 1328 A ` /\ ` identity. (Contr...
falanfal 1329 A ` /\ ` identity. (Contr...
truortru 1330 A ` \/ ` identity. (Contr...
truorfal 1331 A ` \/ ` identity. (Contr...
falortru 1332 A ` \/ ` identity. (Contr...
falorfal 1333 A ` \/ ` identity. (Contr...
truimtru 1334 A ` -> ` identity. (Contr...
truimfal 1335 A ` -> ` identity. (Contr...
falimtru 1336 A ` -> ` identity. (Contr...
falimfal 1337 A ` -> ` identity. (Contr...
nottru 1338 A ` -. ` identity. (Contr...
notfal 1339 A ` -. ` identity. (Contr...
trubitru 1340 A ` <-> ` identity. (Cont...
trubifal 1341 A ` <-> ` identity. (Cont...
falbitru 1342 A ` <-> ` identity. (Cont...
falbifal 1343 A ` <-> ` identity. (Cont...
trunantru 1344 A ` -/\ ` identity. (Cont...
trunanfal 1345 A ` -/\ ` identity. (Cont...
falnantru 1346 A ` -/\ ` identity. (Cont...
falnanfal 1347 A ` -/\ ` identity. (Cont...
truxortru 1348 A ` \/_ ` identity. (Cont...
truxorfal 1349 A ` \/_ ` identity. (Cont...
falxortru 1350 A ` \/_ ` identity. (Cont...
falxorfal 1351 A ` \/_ ` identity. (Cont...
ee22 1352 Virtual deduction rule ~ e...
ee12an 1353 ~ e12an without virtual de...
ee23 1354 ~ e23 without virtual dedu...
exbir 1355 Exportation implication al...
3impexp 1356 ~ impexp with a 3-conjunct...
3impexpbicom 1357 ~ 3impexp with bicondition...
3impexpbicomi 1358 Deduction form of ~ 3impex...
ancomsimp 1359 Closed form of ~ ancoms . ...
exp3acom3r 1360 Export and commute anteced...
exp3acom23g 1361 Implication form of ~ exp3...
exp3acom23 1362 The exportation deduction ...
simplbi2comg 1363 Implication form of ~ simp...
simplbi2com 1364 A deduction eliminating a ...
ee21 1365 ~ e21 without virtual dedu...
ee10 1366 ~ e10 without virtual dedu...
ee02 1367 ~ e02 without virtual dedu...
hadbi123d 1372 Equality theorem for half ...
cadbi123d 1373 Equality theorem for adder...
hadbi123i 1374 Equality theorem for half ...
cadbi123i 1375 Equality theorem for adder...
hadass 1376 Associative law for triple...
hadbi 1377 The half adder is the same...
hadcoma 1378 Commutative law for triple...
hadcomb 1379 Commutative law for triple...
hadrot 1380 Rotation law for triple XO...
cador 1381 Write the adder carry in d...
cadan 1382 Write the adder carry in c...
hadnot 1383 The half adder distributes...
cadnot 1384 The adder carry distribute...
cadcoma 1385 Commutative law for adder ...
cadcomb 1386 Commutative law for adder ...
cadrot 1387 Rotation law for adder car...
cad1 1388 If one parameter is true, ...
cad11 1389 If two parameters are true...
cad0 1390 If one parameter is false,...
cadtru 1391 Rotation law for adder car...
had1 1392 If the first parameter is ...
had0 1393 If the first parameter is ...
meredith 1394 Carew Meredith's sole axio...
axmeredith 1395 Alias for ~ meredith which...
merlem1 1397 Step 3 of Meredith's proof...
merlem2 1398 Step 4 of Meredith's proof...
merlem3 1399 Step 7 of Meredith's proof...
merlem4 1400 Step 8 of Meredith's proof...
merlem5 1401 Step 11 of Meredith's proo...
merlem6 1402 Step 12 of Meredith's proo...
merlem7 1403 Between steps 14 and 15 of...
merlem8 1404 Step 15 of Meredith's proo...
merlem9 1405 Step 18 of Meredith's proo...
merlem10 1406 Step 19 of Meredith's proo...
merlem11 1407 Step 20 of Meredith's proo...
merlem12 1408 Step 28 of Meredith's proo...
merlem13 1409 Step 35 of Meredith's proo...
luk-1 1410 1 of 3 axioms for proposit...
luk-2 1411 2 of 3 axioms for proposit...
luk-3 1412 3 of 3 axioms for proposit...
luklem1 1413 Used to rederive standard ...
luklem2 1414 Used to rederive standard ...
luklem3 1415 Used to rederive standard ...
luklem4 1416 Used to rederive standard ...
luklem5 1417 Used to rederive standard ...
luklem6 1418 Used to rederive standard ...
luklem7 1419 Used to rederive standard ...
luklem8 1420 Used to rederive standard ...
ax1 1421 Standard propositional axi...
ax2 1422 Standard propositional axi...
ax3 1423 Standard propositional axi...
nic-dfim 1424 Define implication in term...
nic-dfneg 1425 Define negation in terms o...
nic-mp 1426 Derive Nicod's rule of mod...
nic-mpALT 1427 A direct proof of ~ nic-mp...
nic-ax 1428 Nicod's axiom derived from...
nic-axALT 1429 A direct proof of ~ nic-ax...
nic-imp 1430 Inference for ~ nic-mp usi...
nic-idlem1 1431 Lemma for ~ nic-id . (Con...
nic-idlem2 1432 Lemma for ~ nic-id . Infe...
nic-id 1433 Theorem ~ id expressed wit...
nic-swap 1434 ` -/\ ` is symmetric. (Co...
nic-isw1 1435 Inference version of ~ nic...
nic-isw2 1436 Inference for swapping nes...
nic-iimp1 1437 Inference version of ~ nic...
nic-iimp2 1438 Inference version of ~ nic...
nic-idel 1439 Inference to remove the tr...
nic-ich 1440 Chained inference. (Contr...
nic-idbl 1441 Double the terms. Since d...
nic-bijust 1442 For nic-* definitions, the...
nic-bi1 1443 Inference to extract one s...
nic-bi2 1444 Inference to extract the o...
nic-stdmp 1445 Derive the standard modus ...
nic-luk1 1446 Proof of ~ luk-1 from ~ ni...
nic-luk2 1447 Proof of ~ luk-2 from ~ ni...
nic-luk3 1448 Proof of ~ luk-3 from ~ ni...
lukshef-ax1 1449 This alternative axiom for...
lukshefth1 1450 Lemma for ~ renicax . (Co...
lukshefth2 1451 Lemma for ~ renicax . (Co...
renicax 1452 A rederivation of ~ nic-ax...
tbw-bijust 1453 Justification for ~ tbw-ne...
tbw-negdf 1454 The definition of negation...
tbw-ax1 1455 The first of four axioms i...
tbw-ax2 1456 The second of four axioms ...
tbw-ax3 1457 The third of four axioms i...
tbw-ax4 1458 The fourth of four axioms ...
tbwsyl 1459 Used to rederive the Lukas...
tbwlem1 1460 Used to rederive the Lukas...
tbwlem2 1461 Used to rederive the Lukas...
tbwlem3 1462 Used to rederive the Lukas...
tbwlem4 1463 Used to rederive the Lukas...
tbwlem5 1464 Used to rederive the Lukas...
re1luk1 1465 ~ luk-1 derived from the T...
re1luk2 1466 ~ luk-2 derived from the T...
re1luk3 1467 ~ luk-3 derived from the T...
merco1 1468 A single axiom for proposi...
merco1lem1 1469 Used to rederive the Tarsk...
retbwax4 1470 ~ tbw-ax4 rederived from ~...
retbwax2 1471 ~ tbw-ax2 rederived from ~...
merco1lem2 1472 Used to rederive the Tarsk...
merco1lem3 1473 Used to rederive the Tarsk...
merco1lem4 1474 Used to rederive the Tarsk...
merco1lem5 1475 Used to rederive the Tarsk...
merco1lem6 1476 Used to rederive the Tarsk...
merco1lem7 1477 Used to rederive the Tarsk...
retbwax3 1478 ~ tbw-ax3 rederived from ~...
merco1lem8 1479 Used to rederive the Tarsk...
merco1lem9 1480 Used to rederive the Tarsk...
merco1lem10 1481 Used to rederive the Tarsk...
merco1lem11 1482 Used to rederive the Tarsk...
merco1lem12 1483 Used to rederive the Tarsk...
merco1lem13 1484 Used to rederive the Tarsk...
merco1lem14 1485 Used to rederive the Tarsk...
merco1lem15 1486 Used to rederive the Tarsk...
merco1lem16 1487 Used to rederive the Tarsk...
merco1lem17 1488 Used to rederive the Tarsk...
merco1lem18 1489 Used to rederive the Tarsk...
retbwax1 1490 ~ tbw-ax1 rederived from ~...
merco2 1491 A single axiom for proposi...
mercolem1 1492 Used to rederive the Tarsk...
mercolem2 1493 Used to rederive the Tarsk...
mercolem3 1494 Used to rederive the Tarsk...
mercolem4 1495 Used to rederive the Tarsk...
mercolem5 1496 Used to rederive the Tarsk...
mercolem6 1497 Used to rederive the Tarsk...
mercolem7 1498 Used to rederive the Tarsk...
mercolem8 1499 Used to rederive the Tarsk...
re1tbw1 1500 ~ tbw-ax1 rederived from ~...
re1tbw2 1501 ~ tbw-ax2 rederived from ~...
re1tbw3 1502 ~ tbw-ax3 rederived from ~...
re1tbw4 1503 ~ tbw-ax4 rederived from ~...
rb-bijust 1504 Justification for ~ rb-imd...
rb-imdf 1505 The definition of implicat...
anmp 1506 Modus ponens for ` \/ ` ` ...
rb-ax1 1507 The first of four axioms i...
rb-ax2 1508 The second of four axioms ...
rb-ax3 1509 The third of four axioms i...
rb-ax4 1510 The fourth of four axioms ...
rbsyl 1511 Used to rederive the Lukas...
rblem1 1512 Used to rederive the Lukas...
rblem2 1513 Used to rederive the Lukas...
rblem3 1514 Used to rederive the Lukas...
rblem4 1515 Used to rederive the Lukas...
rblem5 1516 Used to rederive the Lukas...
rblem6 1517 Used to rederive the Lukas...
rblem7 1518 Used to rederive the Lukas...
re1axmp 1519 ~ ax-mp derived from Russe...
re2luk1 1520 ~ luk-1 derived from Russe...
re2luk2 1521 ~ luk-2 derived from Russe...
re2luk3 1522 ~ luk-3 derived from Russe...
mpto1 1523 Modus ponendo tollens 1, o...
mpto2 1524 Modus ponendo tollens 2, o...
mtp-xor 1525 Modus tollendo ponens (ori...
mtp-xorOLD 1526 Obsolete version of ~ mtp-...
mtp-or 1527 Modus tollendo ponens (inc...
mtp-orOLD 1528 Obsolete version of ~ mtp-...
alnex 1532 Theorem 19.7 of [Margaris]...
gen2 1536 Generalization applied twi...
mpg 1537 Modus ponens combined with...
mpgbi 1538 Modus ponens on biconditio...
mpgbir 1539 Modus ponens on biconditio...
nfi 1540 Deduce that ` x ` is not f...
hbth 1541 No variable is (effectivel...
nfth 1542 No variable is (effectivel...
nftru 1543 The true constant has no f...
nex 1544 Generalization rule for ne...
nfnth 1545 No variable is (effectivel...
alim 1547 Theorem 19.20 of [Margaris...
alimi 1548 Inference quantifying both...
2alimi 1549 Inference doubly quantifyi...
al2imi 1550 Inference quantifying ante...
alanimi 1551 Variant of ~ al2imi with c...
alimdh 1552 Deduction from Theorem 19....
albi 1553 Theorem 19.15 of [Margaris...
alrimih 1554 Inference from Theorem 19....
albii 1555 Inference adding universal...
2albii 1556 Inference adding two unive...
hbxfrbi 1557 A utility lemma to transfe...
nfbii 1558 Equality theorem for not-f...
nfxfr 1559 A utility lemma to transfe...
nfxfrd 1560 A utility lemma to transfe...
alex 1561 Theorem 19.6 of [Margaris]...
2nalexn 1562 Part of theorem *11.5 in [...
exnal 1563 Theorem 19.14 of [Margaris...
exim 1564 Theorem 19.22 of [Margaris...
eximi 1565 Inference adding existenti...
2eximi 1566 Inference adding two exist...
alinexa 1567 A transformation of quanti...
alexn 1568 A relationship between two...
2exnexn 1569 Theorem *11.51 in [Whitehe...
exbi 1570 Theorem 19.18 of [Margaris...
exbii 1571 Inference adding existenti...
2exbii 1572 Inference adding two exist...
3exbii 1573 Inference adding 3 existen...
exanali 1574 A transformation of quanti...
exancom 1575 Commutation of conjunction...
alrimdh 1576 Deduction from Theorem 19....
eximdh 1577 Deduction from Theorem 19....
nexdh 1578 Deduction for generalizati...
albidh 1579 Formula-building rule for ...
exbidh 1580 Formula-building rule for ...
exsimpl 1581 Simplification of an exist...
19.26 1582 Theorem 19.26 of [Margaris...
19.26-2 1583 Theorem 19.26 of [Margaris...
19.26-3an 1584 Theorem 19.26 of [Margaris...
19.29 1585 Theorem 19.29 of [Margaris...
19.29r 1586 Variation of Theorem 19.29...
19.29r2 1587 Variation of Theorem 19.29...
19.29x 1588 Variation of Theorem 19.29...
19.35 1589 Theorem 19.35 of [Margaris...
19.35i 1590 Inference from Theorem 19....
19.35ri 1591 Inference from Theorem 19....
19.25 1592 Theorem 19.25 of [Margaris...
19.30 1593 Theorem 19.30 of [Margaris...
19.43 1594 Theorem 19.43 of [Margaris...
19.43OLD 1595 Obsolete proof of ~ 19.43 ...
19.33 1596 Theorem 19.33 of [Margaris...
19.33b 1597 The antecedent provides a ...
19.40 1598 Theorem 19.40 of [Margaris...
19.40-2 1599 Theorem *11.42 in [Whitehe...
albiim 1600 Split a biconditional and ...
2albiim 1601 Split a biconditional and ...
exintrbi 1602 Add/remove a conjunct in t...
exintr 1603 Introduce a conjunct in th...
alsyl 1604 Theorem *10.3 in [Whitehea...
a17d 1606 ~ ax-17 with antecedent. ...
nfv 1607 If ` x ` is not present in...
nfvd 1608 ~ nfv with antecedent. Us...
alimdv 1609 Deduction from Theorem 19....
eximdv 1610 Deduction from Theorem 19....
2alimdv 1611 Deduction from Theorem 19....
2eximdv 1612 Deduction from Theorem 19....
albidv 1613 Formula-building rule for ...
exbidv 1614 Formula-building rule for ...
2albidv 1615 Formula-building rule for ...
2exbidv 1616 Formula-building rule for ...
3exbidv 1617 Formula-building rule for ...
4exbidv 1618 Formula-building rule for ...
alrimiv 1619 Inference from Theorem 19....
alrimivv 1620 Inference from Theorem 19....
alrimdv 1621 Deduction from Theorem 19....
nfdv 1622 Apply the definition of no...
2ax17 1623 Quantification of two vari...
weq 1626 Extend wff definition to i...
equs3 1627 Lemma used in proofs of su...
speimfw 1628 Specialization, with addit...
spimfw 1629 Specialization, with addit...
ax11i 1630 Inference that has ~ ax-11...
sbequ2 1633 An equality theorem for su...
sb1 1634 One direction of a simplif...
sbimi 1635 Infer substitution into an...
sbbii 1636 Infer substitution into bo...
ax9v 1638 Axiom B7 of [Tarski] p. 75...
a9ev 1639 At least one individual ex...
spimw 1640 Specialization. Lemma 8 o...
spimvw 1641 Specialization. Lemma 8 o...
spnfw 1642 Weak version of ~ sp . Us...
cbvaliw 1643 Change bound variable. Us...
cbvalivw 1644 Change bound variable. Us...
equid 1646 Identity law for equality....
nfequid 1647 Bound-variable hypothesis ...
equcomi 1648 Commutative law for equali...
equcom 1649 Commutative law for equali...
equequ1 1650 An equivalence law for equ...
equequ2 1651 An equivalence law for equ...
stdpc6 1652 One of the two equality ax...
equcoms 1653 An inference commuting equ...
equtr 1654 A transitive law for equal...
equtrr 1655 A transitive law for equal...
equtr2 1656 A transitive law for equal...
ax12b 1657 Two equivalent ways of exp...
ax12bOLD 1658 Obsolete version of ~ ax12...
spfw 1659 Weak version of ~ sp . Us...
spnfwOLD 1660 Weak version of ~ sp . Us...
19.8w 1661 Weak version of ~ 19.8a . ...
spw 1662 Weak version of specializa...
spvw 1663 Version of ~ sp when ` x `...
19.3v 1664 Special case of Theorem 19...
19.9v 1665 Special case of Theorem 19...
exlimdv 1666 Deduction from Theorem 19....
exlimddv 1667 Existential elimination ru...
exlimiv 1668 Inference from Theorem 19....
exlimivv 1669 Inference from Theorem 19....
exlimdvv 1670 Deduction from Theorem 19....
sptruw 1671 Version of ~ sp when ` ph ...
spfalw 1672 Version of ~ sp when ` ph ...
19.2 1673 Theorem 19.2 of [Margaris]...
19.39 1674 Theorem 19.39 of [Margaris...
19.24 1675 Theorem 19.24 of [Margaris...
19.34 1676 Theorem 19.34 of [Margaris...
cbvalw 1677 Change bound variable. Us...
cbvalvw 1678 Change bound variable. Us...
cbvexvw 1679 Change bound variable. Us...
alcomiw 1680 Weak version of ~ alcom . ...
hbn1fw 1681 Weak version of ~ ax-6 fro...
hbn1w 1682 Weak version of ~ hbn1 . ...
hba1w 1683 Weak version of ~ hba1 . ...
hbe1w 1684 Weak version of ~ hbe1 . ...
hbalw 1685 Weak version of ~ hbal . ...
wel 1687 Extend wff definition to i...
elequ1 1689 An identity law for the no...
elequ2 1691 An identity law for the no...
ax9dgen 1692 Tarski's system uses the w...
ax6w 1693 Weak version of ~ ax-6 fro...
ax7w 1694 Weak version of ~ ax-7 fro...
ax7dgen 1695 Degenerate instance of ~ a...
ax11wlem 1696 Lemma for weak version of ...
ax11w 1697 Weak version of ~ ax-11 fr...
ax11dgen 1698 Degenerate instance of ~ a...
ax11wdemo 1699 Example of an application ...
ax12w 1700 Weak version (principal in...
ax12dgen1 1701 Degenerate instance of ~ a...
ax12dgen2 1702 Degenerate instance of ~ a...
ax12dgen3 1703 Degenerate instance of ~ a...
ax12dgen4 1704 Degenerate instance of ~ a...
hbn1 1706 ` x ` is not free in ` -. ...
hbe1 1707 ` x ` is not free in ` E. ...
nfe1 1708 ` x ` is not free in ` E. ...
modal-5 1709 The analog in our "pure" p...
a7s 1711 Swap quantifiers in an ant...
hbal 1712 If ` x ` is not free in ` ...
alcom 1713 Theorem 19.5 of [Margaris]...
alrot3 1714 Theorem *11.21 in [Whitehe...
alrot4 1715 Rotate 4 universal quantif...
hbald 1716 Deduction form of bound-va...
sp 1718 Specialization. A univers...
ax5o 1719 Show that the original axi...
19.8a 1720 If a wff is true, it is tr...
hba1 1721 ` x ` is not free in ` A. ...
hbn 1722 If ` x ` is not free in ` ...
hbimd 1723 Deduction form of bound-va...
spimeh 1724 Existential introduction, ...
ax6o 1725 Show that the original axi...
hbnt 1726 Closed theorem version of ...
hbim 1727 If ` x ` is not free in ` ...
19.9ht 1728 A closed version of ~ 19.9...
19.9h 1729 A wff may be existentially...
19.23h 1730 Theorem 19.23 of [Margaris...
exlimih 1731 Inference from Theorem 19....
equsalhw 1732 Weaker version of ~ equsal...
19.21h 1733 Theorem 19.21 of [Margaris...
hbim1 1734 A closed form of ~ hbim . ...
hbex 1735 If ` x ` is not free in ` ...
19.12 1736 Theorem 19.12 of [Margaris...
dvelimhw 1737 Proof of ~ dvelimh without...
hban 1738 If ` x ` is not free in ` ...
cbv3hv 1739 Lemma for ~ ax10 . Simila...
spi 1740 Inference rule reversing g...
sps 1741 Generalization of antecede...
spsd 1742 Deduction generalizing ant...
nfr 1743 Consequence of the definit...
nfri 1744 Consequence of the definit...
nfrd 1745 Consequence of the definit...
alimd 1746 Deduction from Theorem 19....
alrimi 1747 Inference from Theorem 19....
nfd 1748 Deduce that ` x ` is not f...
nfdh 1749 Deduce that ` x ` is not f...
alrimdd 1750 Deduction from Theorem 19....
alrimd 1751 Deduction from Theorem 19....
eximd 1752 Deduction from Theorem 19....
nexd 1753 Deduction for generalizati...
albid 1754 Formula-building rule for ...
exbid 1755 Formula-building rule for ...
nfbidf 1756 An equality theorem for ef...
a6e 1757 Abbreviated version of ~ a...
nfa1 1758 ` x ` is not free in ` A. ...
nfnf1 1759 ` x ` is not free in ` F/ ...
a5i 1760 Inference version of ~ ax5...
hb3an 1761 If ` x ` is not free in ` ...
nfnd 1762 If ` x ` is not free in ` ...
nfimd 1763 If ` x ` is not free in ` ...
nfbid 1764 If ` x ` is not free in ` ...
nfand 1765 If ` x ` is not free in ` ...
nf3and 1766 Deduction form of bound-va...
nfn 1767 If ` x ` is not free in ` ...
nfal 1768 If ` x ` is not free in ` ...
nfex 1769 If ` x ` is not free in ` ...
nfnf 1770 If ` x ` is not free in ` ...
nfim 1771 If ` x ` is not free in ` ...
nfor 1772 If ` x ` is not free in ` ...
nfan 1773 If ` x ` is not free in ` ...
nfbi 1774 If ` x ` is not free in ` ...
nf3or 1775 If ` x ` is not free in ` ...
nf3an 1776 If ` x ` is not free in ` ...
nfald 1777 If ` x ` is not free in ` ...
nfexd 1778 If ` x ` is not free in ` ...
nfa2 1779 Lemma 24 of [Monk2] p. 114...
nfia1 1780 Lemma 23 of [Monk2] p. 114...
modal-b 1781 The analog in our "pure" p...
19.2g 1782 Theorem 19.2 of [Margaris]...
19.3 1783 A wff may be quantified wi...
19.9t 1784 A closed version of ~ 19.9...
19.9 1785 A wff may be existentially...
19.9d 1786 A deduction version of one...
excomim 1787 One direction of Theorem 1...
excom 1788 Theorem 19.11 of [Margaris...
19.16 1789 Theorem 19.16 of [Margaris...
19.17 1790 Theorem 19.17 of [Margaris...
19.19 1791 Theorem 19.19 of [Margaris...
19.21t 1792 Closed form of Theorem 19....
19.21 1793 Theorem 19.21 of [Margaris...
19.21-2 1794 Theorem 19.21 of [Margaris...
stdpc5 1795 An axiom scheme of standar...
19.21bi 1796 Inference from Theorem 19....
19.21bbi 1797 Inference removing double ...
19.23t 1798 Closed form of Theorem 19....
19.23 1799 Theorem 19.23 of [Margaris...
nf2 1800 An alternative definition ...
nf3 1801 An alternative definition ...
nf4 1802 Variable ` x ` is effectiv...
exlimi 1803 Inference from Theorem 19....
19.23bi 1804 Inference from Theorem 19....
exlimd 1805 Deduction from Theorem 19....
exlimdh 1806 Deduction from Theorem 19....
19.27 1807 Theorem 19.27 of [Margaris...
19.28 1808 Theorem 19.28 of [Margaris...
19.36 1809 Theorem 19.36 of [Margaris...
19.36i 1810 Inference from Theorem 19....
19.37 1811 Theorem 19.37 of [Margaris...
19.38 1812 Theorem 19.38 of [Margaris...
19.32 1813 Theorem 19.32 of [Margaris...
19.31 1814 Theorem 19.31 of [Margaris...
19.44 1815 Theorem 19.44 of [Margaris...
19.45 1816 Theorem 19.45 of [Margaris...
19.41 1817 Theorem 19.41 of [Margaris...
19.42 1818 Theorem 19.42 of [Margaris...
excom13 1819 Swap 1st and 3rd existenti...
exrot3 1820 Rotate existential quantif...
exrot4 1821 Rotate existential quantif...
nexr 1822 Inference from ~ 19.8a . ...
nfim1 1823 A closed form of ~ nfim . ...
nfan1 1824 A closed form of ~ nfan . ...
exan 1825 Place a conjunct in the sc...
hbnd 1826 Deduction form of bound-va...
aaan 1827 Rearrange universal quanti...
eeor 1828 Rearrange existential quan...
qexmid 1829 Quantified "excluded middl...
equs5a 1830 A property related to subs...
equs5e 1831 A property related to subs...
exlimdd 1832 Existential elimination ru...
19.21v 1833 Special case of Theorem 19...
19.23v 1834 Special case of Theorem 19...
19.23vv 1835 Theorem 19.23 of [Margaris...
pm11.53 1836 Theorem *11.53 in [Whitehe...
19.27v 1837 Theorem 19.27 of [Margaris...
19.28v 1838 Theorem 19.28 of [Margaris...
19.36v 1839 Special case of Theorem 19...
19.36aiv 1840 Inference from Theorem 19....
19.12vv 1841 Special case of ~ 19.12 wh...
19.37v 1842 Special case of Theorem 19...
19.37aiv 1843 Inference from Theorem 19....
19.41v 1844 Special case of Theorem 19...
19.41vv 1845 Theorem 19.41 of [Margaris...
19.41vvv 1846 Theorem 19.41 of [Margaris...
19.41vvvv 1847 Theorem 19.41 of [Margaris...
19.42v 1848 Special case of Theorem 19...
exdistr 1849 Distribution of existentia...
19.42vv 1850 Theorem 19.42 of [Margaris...
19.42vvv 1851 Theorem 19.42 of [Margaris...
exdistr2 1852 Distribution of existentia...
3exdistr 1853 Distribution of existentia...
4exdistr 1854 Distribution of existentia...
eean 1855 Rearrange existential quan...
eeanv 1856 Rearrange existential quan...
eeeanv 1857 Rearrange existential quan...
ee4anv 1858 Rearrange existential quan...
nexdv 1859 Deduction for generalizati...
stdpc7 1860 One of the two equality ax...
sbequ1 1861 An equality theorem for su...
sbequ12 1862 An equality theorem for su...
sbequ12r 1863 An equality theorem for su...
sbequ12a 1864 An equality theorem for su...
sbid 1865 An identity theorem for su...
sb4a 1866 A version of ~ sb4 that do...
sb4e 1867 One direction of a simplif...
ax12v 1869 A weaker version of ~ ax-1...
ax12olem1 1870 Lemma for ~ ax12o . Simil...
ax12olem2 1871 Lemma for ~ ax12o . Negat...
ax12olem3 1872 Lemma for ~ ax12o . Show ...
ax12olem4 1873 Lemma for ~ ax12o . Const...
ax12olem5 1874 Lemma for ~ ax12o . See ~...
ax12olem6 1875 Lemma for ~ ax12o . Deriv...
ax12olem7 1876 Lemma for ~ ax12o . Deriv...
ax12o 1877 Derive set.mm's original ~...
ax10lem1 1878 Lemma for ~ ax10 . Change...
ax10lem2 1879 Lemma for ~ ax10 . Change...
ax10lem3 1880 Lemma for ~ ax10 . Simila...
dvelimv 1881 Similar to ~ dvelim with f...
dveeq2 1882 Quantifier introduction wh...
ax10lem4 1883 Lemma for ~ ax10 . Change...
ax10lem5 1884 Lemma for ~ ax10 . Change...
ax10lem6 1885 Lemma for ~ ax10 . Simila...
ax10 1886 Derive set.mm's original ~...
a16g 1887 Generalization of ~ ax16 ....
aecom 1888 Commutation law for identi...
aecoms 1889 A commutation rule for ide...
naecoms 1890 A commutation rule for dis...
ax9 1891 Theorem showing that ~ ax-...
ax9o 1892 Show that the original axi...
a9e 1893 At least one individual ex...
ax10o 1894 Show that ~ ax-10o can be ...
hbae 1895 All variables are effectiv...
nfae 1896 All variables are effectiv...
hbnae 1897 All variables are effectiv...
nfnae 1898 All variables are effectiv...
hbnaes 1899 Rule that applies ~ hbnae ...
nfeqf 1900 A variable is effectively ...
equs4 1901 Lemma used in proofs of su...
equsal 1902 A useful equivalence relat...
equsalh 1903 A useful equivalence relat...
equsex 1904 A useful equivalence relat...
equsexh 1905 A useful equivalence relat...
dvelimh 1906 Version of ~ dvelim withou...
dral1 1907 Formula-building lemma for...
dral2 1908 Formula-building lemma for...
drex1 1909 Formula-building lemma for...
drex2 1910 Formula-building lemma for...
drnf1 1911 Formula-building lemma for...
drnf2 1912 Formula-building lemma for...
exdistrf 1913 Distribution of existentia...
nfald2 1914 Variation on ~ nfald which...
nfexd2 1915 Variation on ~ nfexd which...
spimt 1916 Closed theorem form of ~ s...
spim 1917 Specialization, using impl...
spime 1918 Existential introduction, ...
spimed 1919 Deduction version of ~ spi...
cbv1h 1920 Rule used to change bound ...
cbv1 1921 Rule used to change bound ...
cbv2h 1922 Rule used to change bound ...
cbv2 1923 Rule used to change bound ...
cbv3 1924 Rule used to change bound ...
cbv3h 1925 Rule used to change bound ...
cbval 1926 Rule used to change bound ...
cbvex 1927 Rule used to change bound ...
chvar 1928 Implicit substitution of `...
equvini 1929 A variable introduction la...
equveli 1930 A variable elimination law...
equs45f 1931 Two ways of expressing sub...
spimv 1932 A version of ~ spim with a...
aev 1933 A "distinctor elimination"...
ax11v2 1934 Recovery of ~ ax-11o from ...
ax11a2 1935 Derive ~ ax-11o from a hyp...
ax11o 1936 Derivation of set.mm's ori...
ax11b 1937 A bidirectional version of...
equs5 1938 Lemma used in proofs of su...
dvelimf 1939 Version of ~ dvelimv witho...
spv 1940 Specialization, using impl...
spimev 1941 Distinct-variable version ...
speiv 1942 Inference from existential...
equvin 1943 A variable introduction la...
cbvalv 1944 Rule used to change bound ...
cbvexv 1945 Rule used to change bound ...
cbval2 1946 Rule used to change bound ...
cbvex2 1947 Rule used to change bound ...
cbval2v 1948 Rule used to change bound ...
cbvex2v 1949 Rule used to change bound ...
cbvald 1950 Deduction used to change b...
cbvexd 1951 Deduction used to change b...
cbvaldva 1952 Rule used to change the bo...
cbvexdva 1953 Rule used to change the bo...
cbvex4v 1954 Rule used to change bound ...
chvarv 1955 Implicit substitution of `...
cleljust 1956 When the class variables i...
cleljustALT 1957 When the class variables i...
dvelim 1958 This theorem can be used t...
dvelimnf 1959 Version of ~ dvelim using ...
dveeq1 1960 Quantifier introduction wh...
dveel1 1961 Quantifier introduction wh...
dveel2 1962 Quantifier introduction wh...
ax15 1963 Axiom ~ ax-15 is redundant...
drsb1 1964 Formula-building lemma for...
sb2 1965 One direction of a simplif...
stdpc4 1966 The specialization axiom o...
sbft 1967 Substitution has no effect...
sbf 1968 Substitution for a variabl...
sbh 1969 Substitution for a variabl...
sbf2 1970 Substitution has no effect...
sb6x 1971 Equivalence involving subs...
nfs1f 1972 If ` x ` is not free in ` ...
sbequ5 1973 Substitution does not chan...
sbequ6 1974 Substitution does not chan...
sbt 1975 A substitution into a theo...
equsb1 1976 Substitution applied to an...
equsb2 1977 Substitution applied to an...
sbied 1978 Conversion of implicit sub...
sbiedv 1979 Conversion of implicit sub...
sbie 1980 Conversion of implicit sub...
sb6f 1981 Equivalence for substituti...
sb5f 1982 Equivalence for substituti...
hbsb2a 1983 Special case of a bound-va...
hbsb2e 1984 Special case of a bound-va...
hbsb3 1985 If ` y ` is not free in ` ...
nfs1 1986 If ` y ` is not free in ` ...
ax16 1987 Proof of older axiom ~ ax-...
ax16i 1988 Inference with ~ ax16 as i...
ax16ALT 1989 Alternate proof of ~ ax16 ...
ax16ALT2 1990 Alternate proof of ~ ax16 ...
a16gALT 1991 A generalization of axiom ...
a16gb 1992 A generalization of axiom ...
a16nf 1993 If ~ dtru is false, then t...
sb3 1994 One direction of a simplif...
sb4 1995 One direction of a simplif...
sb4b 1996 Simplified definition of s...
dfsb2 1997 An alternate definition of...
dfsb3 1998 An alternate definition of...
hbsb2 1999 Bound-variable hypothesis ...
nfsb2 2000 Bound-variable hypothesis ...
sbequi 2001 An equality theorem for su...
sbequ 2002 An equality theorem for su...
drsb2 2003 Formula-building lemma for...
sbn 2004 Negation inside and outsid...
sbi1 2005 Removal of implication fro...
sbi2 2006 Introduction of implicatio...
sbim 2007 Implication inside and out...
sbor 2008 Logical OR inside and outs...
sbrim 2009 Substitution with a variab...
sblim 2010 Substitution with a variab...
sban 2011 Conjunction inside and out...
sb3an 2012 Conjunction inside and out...
sbbi 2013 Equivalence inside and out...
sblbis 2014 Introduce left bicondition...
sbrbis 2015 Introduce right biconditio...
sbrbif 2016 Introduce right biconditio...
spsbe 2017 A specialization theorem. ...
spsbim 2018 Specialization of implicat...
spsbbi 2019 Specialization of bicondit...
sbbid 2020 Deduction substituting bot...
sbequ8 2021 Elimination of equality fr...
nfsb4t 2022 A variable not free remain...
nfsb4 2023 A variable not free remain...
dvelimdf 2024 Deduction form of ~ dvelim...
sbco 2025 A composition law for subs...
sbid2 2026 An identity law for substi...
sbidm 2027 An idempotent law for subs...
sbco2 2028 A composition law for subs...
sbco2d 2029 A composition law for subs...
sbco3 2030 A composition law for subs...
sbcom 2031 A commutativity law for su...
sb5rf 2032 Reversed substitution. (C...
sb6rf 2033 Reversed substitution. (C...
sb8 2034 Substitution of variable i...
sb8e 2035 Substitution of variable i...
sb9i 2036 Commutation of quantificat...
sb9 2037 Commutation of quantificat...
ax11v 2038 This is a version of ~ ax-...
sb56 2039 Two equivalent ways of exp...
sb6 2040 Equivalence for substituti...
sb5 2041 Equivalence for substituti...
equsb3lem 2042 Lemma for ~ equsb3 . (Con...
equsb3 2043 Substitution applied to an...
elsb3 2044 Substitution applied to an...
elsb4 2045 Substitution applied to an...
hbs1 2046 ` x ` is not free in ` [ y...
nfs1v 2047 ` x ` is not free in ` [ y...
sbhb 2048 Two ways of expressing " `...
sbnf2 2049 Two ways of expressing " `...
nfsb 2050 If ` z ` is not free in ` ...
hbsb 2051 If ` z ` is not free in ` ...
nfsbd 2052 Deduction version of ~ nfs...
2sb5 2053 Equivalence for double sub...
2sb6 2054 Equivalence for double sub...
sbcom2 2055 Commutativity law for subs...
pm11.07 2056 Theorem *11.07 in [Whitehe...
sb6a 2057 Equivalence for substituti...
2sb5rf 2058 Reversed double substituti...
2sb6rf 2059 Reversed double substituti...
dfsb7 2060 An alternate definition of...
sb7f 2061 This version of ~ dfsb7 do...
sb7h 2062 This version of ~ dfsb7 do...
sb10f 2063 Hao Wang's identity axiom ...
sbid2v 2064 An identity law for substi...
sbelx 2065 Elimination of substitutio...
sbel2x 2066 Elimination of double subs...
sbal1 2067 A theorem used in eliminat...
sbal 2068 Move universal quantifier ...
sbex 2069 Move existential quantifie...
sbalv 2070 Quantify with new variable...
exsb 2071 An equivalent expression f...
exsbOLD 2072 An equivalent expression f...
2exsb 2073 An equivalent expression f...
dvelimALT 2074 Version of ~ dvelim that d...
sbal2 2075 Move quantifier in and out...
ax4 2086 This theorem repeats ~ sp ...
ax5 2087 Rederivation of axiom ~ ax...
ax6 2088 Rederivation of axiom ~ ax...
ax9from9o 2089 Rederivation of axiom ~ ax...
hba1-o 2090 ` x ` is not free in ` A. ...
a5i-o 2091 Inference version of ~ ax-...
aecom-o 2092 Commutation law for identi...
aecoms-o 2093 A commutation rule for ide...
hbae-o 2094 All variables are effectiv...
dral1-o 2095 Formula-building lemma for...
ax11 2096 Rederivation of axiom ~ ax...
ax12 2097 Derive ~ ax-12 from ~ ax-1...
ax17o 2098 Axiom to quantify a variab...
equid1 2099 Identity law for equality ...
sps-o 2100 Generalization of antecede...
hbequid 2101 Bound-variable hypothesis ...
nfequid-o 2102 Bound-variable hypothesis ...
ax46 2103 Proof of a single axiom th...
ax46to4 2104 Re-derivation of ~ ax-4 fr...
ax46to6 2105 Re-derivation of ~ ax-6o f...
ax67 2106 Proof of a single axiom th...
nfa1-o 2107 ` x ` is not free in ` A. ...
ax67to6 2108 Re-derivation of ~ ax-6o f...
ax67to7 2109 Re-derivation of ~ ax-7 fr...
ax467 2110 Proof of a single axiom th...
ax467to4 2111 Re-derivation of ~ ax-4 fr...
ax467to6 2112 Re-derivation of ~ ax-6o f...
ax467to7 2113 Re-derivation of ~ ax-7 fr...
equidqe 2114 ~ equid with existential q...
ax4sp1 2115 A special case of ~ ax-4 w...
equidq 2116 ~ equid with universal qua...
equid1ALT 2117 Identity law for equality ...
ax10from10o 2118 Rederivation of ~ ax-10 fr...
naecoms-o 2119 A commutation rule for dis...
hbnae-o 2120 All variables are effectiv...
dvelimf-o 2121 Proof of ~ dvelimh that us...
dral2-o 2122 Formula-building lemma for...
aev-o 2123 A "distinctor elimination"...
ax17eq 2124 Theorem to add distinct qu...
dveeq2-o 2125 Quantifier introduction wh...
dveeq2-o16 2126 Version of ~ dveeq2 using ...
a16g-o 2127 A generalization of axiom ...
dveeq1-o 2128 Quantifier introduction wh...
dveeq1-o16 2129 Version of ~ dveeq1 using ...
ax17el 2130 Theorem to add distinct qu...
ax10-16 2131 This theorem shows that, g...
dveel2ALT 2132 Version of ~ dveel2 using ...
ax11f 2133 Basis step for constructin...
ax11eq 2134 Basis step for constructin...
ax11el 2135 Basis step for constructin...
ax11indn 2136 Induction step for constru...
ax11indi 2137 Induction step for constru...
ax11indalem 2138 Lemma for ~ ax11inda2 and ...
ax11inda2ALT 2139 A proof of ~ ax11inda2 tha...
ax11inda2 2140 Induction step for constru...
ax11inda 2141 Induction step for constru...
ax11v2-o 2142 Recovery of ~ ax-11o from ...
ax11a2-o 2143 Derive ~ ax-11o from a hyp...
ax10o-o 2144 Show that ~ ax-10o can be ...
eujust 2147 A soundness justification ...
eujustALT 2148 A soundness justification ...
euf 2151 A version of the existenti...
eubid 2152 Formula-building rule for ...
eubidv 2153 Formula-building rule for ...
eubii 2154 Introduce uniqueness quant...
nfeu1 2155 Bound-variable hypothesis ...
nfmo1 2156 Bound-variable hypothesis ...
nfeud2 2157 Bound-variable hypothesis ...
nfmod2 2158 Bound-variable hypothesis ...
nfeud 2159 Deduction version of ~ nfe...
nfmod 2160 Bound-variable hypothesis ...
nfeu 2161 Bound-variable hypothesis ...
nfmo 2162 Bound-variable hypothesis ...
sb8eu 2163 Variable substitution in u...
sb8mo 2164 Variable substitution in u...
cbveu 2165 Rule used to change bound ...
eu1 2166 An alternate way to expres...
mo 2167 Equivalent definitions of ...
euex 2168 Existential uniqueness imp...
eumo0 2169 Existential uniqueness imp...
eu2 2170 An alternate way of defini...
eu3 2171 An alternate way to expres...
euor 2172 Introduce a disjunct into ...
euorv 2173 Introduce a disjunct into ...
mo2 2174 Alternate definition of "a...
sbmo 2175 Substitution into "at most...
mo3 2176 Alternate definition of "a...
mo4f 2177 "At most one" expressed us...
mo4 2178 "At most one" expressed us...
mobid 2179 Formula-building rule for ...
mobidv 2180 Formula-building rule for ...
mobii 2181 Formula-building rule for ...
cbvmo 2182 Rule used to change bound ...
eu5 2183 Uniqueness in terms of "at...
eu4 2184 Uniqueness using implicit ...
eumo 2185 Existential uniqueness imp...
eumoi 2186 "At most one" inferred fro...
exmoeu 2187 Existence in terms of "at ...
exmoeu2 2188 Existence implies "at most...
moabs 2189 Absorption of existence co...
exmo 2190 Something exists or at mos...
moim 2191 "At most one" is preserved...
moimi 2192 "At most one" is preserved...
morimv 2193 Move antecedent outside of...
euimmo 2194 Uniqueness implies "at mos...
euim 2195 Add existential uniqueness...
moan 2196 "At most one" is still the...
moani 2197 "At most one" is still tru...
moor 2198 "At most one" is still the...
mooran1 2199 "At most one" imports disj...
mooran2 2200 "At most one" exports disj...
moanim 2201 Introduction of a conjunct...
euan 2202 Introduction of a conjunct...
moanimv 2203 Introduction of a conjunct...
moaneu 2204 Nested "at most one" and u...
moanmo 2205 Nested "at most one" quant...
euanv 2206 Introduction of a conjunct...
mopick 2207 "At most one" picks a vari...
eupick 2208 Existential uniqueness "pi...
eupicka 2209 Version of ~ eupick with c...
eupickb 2210 Existential uniqueness "pi...
eupickbi 2211 Theorem *14.26 in [Whitehe...
mopick2 2212 "At most one" can show the...
euor2 2213 Introduce or eliminate a d...
moexex 2214 "At most one" double quant...
moexexv 2215 "At most one" double quant...
2moex 2216 Double quantification with...
2euex 2217 Double quantification with...
2eumo 2218 Double quantification with...
2eu2ex 2219 Double existential uniquen...
2moswap 2220 A condition allowing swap ...
2euswap 2221 A condition allowing swap ...
2exeu 2222 Double existential uniquen...
2mo 2223 Two equivalent expressions...
2mos 2224 Double "exists at most one...
2eu1 2225 Double existential uniquen...
2eu2 2226 Double existential uniquen...
2eu3 2227 Double existential uniquen...
2eu4 2228 This theorem provides us w...
2eu5 2229 An alternate definition of...
2eu6 2230 Two equivalent expressions...
2eu7 2231 Two equivalent expressions...
2eu8 2232 Two equivalent expressions...
euequ1 2233 Equality has existential u...
exists1 2234 Two ways to express "only ...
exists2 2235 A condition implying that ...
barbara 2242 "Barbara", one of the fund...
celarent 2243 "Celarent", one of the syl...
darii 2244 "Darii", one of the syllog...
ferio 2245 "Ferio" ("Ferioque"), one ...
barbari 2246 "Barbari", one of the syll...
celaront 2247 "Celaront", one of the syl...
cesare 2248 "Cesare", one of the syllo...
camestres 2249 "Camestres", one of the sy...
festino 2250 "Festino", one of the syll...
baroco 2251 "Baroco", one of the syllo...
cesaro 2252 "Cesaro", one of the syllo...
camestros 2253 "Camestros", one of the sy...
datisi 2254 "Datisi", one of the syllo...
disamis 2255 "Disamis", one of the syll...
ferison 2256 "Ferison", one of the syll...
bocardo 2257 "Bocardo", one of the syll...
felapton 2258 "Felapton", one of the syl...
darapti 2259 "Darapti", one of the syll...
calemes 2260 "Calemes", one of the syll...
dimatis 2261 "Dimatis", one of the syll...
fresison 2262 "Fresison", one of the syl...
calemos 2263 "Calemos", one of the syll...
fesapo 2264 "Fesapo", one of the syllo...
bamalip 2265 "Bamalip", one of the syll...
axext2 2267 The Axiom of Extensionalit...
axext3 2268 A generalization of the Ax...
axext4 2269 A bidirectional version of...
bm1.1 2270 Any set defined by a prope...
abid 2273 Simplification of class ab...
hbab1 2274 Bound-variable hypothesis ...
nfsab1 2275 Bound-variable hypothesis ...
hbab 2276 Bound-variable hypothesis ...
nfsab 2277 Bound-variable hypothesis ...
dfcleq 2279 The same as ~ df-cleq with...
cvjust 2280 Every set is a class. Pro...
eqriv 2282 Infer equality of classes ...
eqrdv 2283 Deduce equality of classes...
eqrdav 2284 Deduce equality of classes...
eqid 2285 Law of identity (reflexivi...
eqidd 2286 Class identity law with an...
eqcom 2287 Commutative law for class ...
eqcoms 2288 Inference applying commuta...
eqcomi 2289 Inference from commutative...
eqcomd 2290 Deduction from commutative...
eqeq1 2291 Equality implies equivalen...
eqeq1i 2292 Inference from equality to...
eqeq1d 2293 Deduction from equality to...
eqeq2 2294 Equality implies equivalen...
eqeq2i 2295 Inference from equality to...
eqeq2d 2296 Deduction from equality to...
eqeq12 2297 Equality relationship amon...
eqeq12i 2298 A useful inference for sub...
eqeq12d 2299 A useful inference for sub...
eqeqan12d 2300 A useful inference for sub...
eqeqan12rd 2301 A useful inference for sub...
eqtr 2302 Transitive law for class e...
eqtr2 2303 A transitive law for class...
eqtr3 2304 A transitive law for class...
eqtri 2305 An equality transitivity i...
eqtr2i 2306 An equality transitivity i...
eqtr3i 2307 An equality transitivity i...
eqtr4i 2308 An equality transitivity i...
3eqtri 2309 An inference from three ch...
3eqtrri 2310 An inference from three ch...
3eqtr2i 2311 An inference from three ch...
3eqtr2ri 2312 An inference from three ch...
3eqtr3i 2313 An inference from three ch...
3eqtr3ri 2314 An inference from three ch...
3eqtr4i 2315 An inference from three ch...
3eqtr4ri 2316 An inference from three ch...
eqtrd 2317 An equality transitivity d...
eqtr2d 2318 An equality transitivity d...
eqtr3d 2319 An equality transitivity e...
eqtr4d 2320 An equality transitivity e...
3eqtrd 2321 A deduction from three cha...
3eqtrrd 2322 A deduction from three cha...
3eqtr2d 2323 A deduction from three cha...
3eqtr2rd 2324 A deduction from three cha...
3eqtr3d 2325 A deduction from three cha...
3eqtr3rd 2326 A deduction from three cha...
3eqtr4d 2327 A deduction from three cha...
3eqtr4rd 2328 A deduction from three cha...
syl5eq 2329 An equality transitivity d...
syl5req 2330 An equality transitivity d...
syl5eqr 2331 An equality transitivity d...
syl5reqr 2332 An equality transitivity d...
syl6eq 2333 An equality transitivity d...
syl6req 2334 An equality transitivity d...
syl6eqr 2335 An equality transitivity d...
syl6reqr 2336 An equality transitivity d...
sylan9eq 2337 An equality transitivity d...
sylan9req 2338 An equality transitivity d...
sylan9eqr 2339 An equality transitivity d...
3eqtr3g 2340 A chained equality inferen...
3eqtr3a 2341 A chained equality inferen...
3eqtr4g 2342 A chained equality inferen...
3eqtr4a 2343 A chained equality inferen...
eq2tri 2344 A compound transitive infe...
eleq1 2345 Equality implies equivalen...
eleq2 2346 Equality implies equivalen...
eleq12 2347 Equality implies equivalen...
eleq1i 2348 Inference from equality to...
eleq2i 2349 Inference from equality to...
eleq12i 2350 Inference from equality to...
eleq1d 2351 Deduction from equality to...
eleq2d 2352 Deduction from equality to...
eleq12d 2353 Deduction from equality to...
eleq1a 2354 A transitive-type law rela...
eqeltri 2355 Substitution of equal clas...
eqeltrri 2356 Substitution of equal clas...
eleqtri 2357 Substitution of equal clas...
eleqtrri 2358 Substitution of equal clas...
eqeltrd 2359 Substitution of equal clas...
eqeltrrd 2360 Deduction that substitutes...
eleqtrd 2361 Deduction that substitutes...
eleqtrrd 2362 Deduction that substitutes...
3eltr3i 2363 Substitution of equal clas...
3eltr4i 2364 Substitution of equal clas...
3eltr3d 2365 Substitution of equal clas...
3eltr4d 2366 Substitution of equal clas...
3eltr3g 2367 Substitution of equal clas...
3eltr4g 2368 Substitution of equal clas...
syl5eqel 2369 B membership and equality ...
syl5eqelr 2370 B membership and equality ...
syl5eleq 2371 B membership and equality ...
syl5eleqr 2372 B membership and equality ...
syl6eqel 2373 A membership and equality ...
syl6eqelr 2374 A membership and equality ...
syl6eleq 2375 A membership and equality ...
syl6eleqr 2376 A membership and equality ...
eleq2s 2377 Substitution of equal clas...
eqneltrd 2378 If a class is not an eleme...
eqneltrrd 2379 If a class is not an eleme...
neleqtrd 2380 If a class is not an eleme...
neleqtrrd 2381 If a class is not an eleme...
cleqh 2382 Establish equality between...
nelneq 2383 A way of showing two class...
nelneq2 2384 A way of showing two class...
eqsb3lem 2385 Lemma for ~ eqsb3 . (Cont...
eqsb3 2386 Substitution applied to an...
clelsb3 2387 Substitution applied to an...
hbxfreq 2388 A utility lemma to transfe...
hblem 2389 Change the free variable o...
abeq2 2390 Equality of a class variab...
abeq1 2391 Equality of a class variab...
abeq2i 2392 Equality of a class variab...
abeq1i 2393 Equality of a class variab...
abeq2d 2394 Equality of a class variab...
abbi 2395 Equivalent wff's correspon...
abbi2i 2396 Equality of a class variab...
abbii 2397 Equivalent wff's yield equ...
abbid 2398 Equivalent wff's yield equ...
abbidv 2399 Equivalent wff's yield equ...
abbi2dv 2400 Deduction from a wff to a ...
abbi1dv 2401 Deduction from a wff to a ...
abid2 2402 A simplification of class ...
cbvab 2403 Rule used to change bound ...
cbvabv 2404 Rule used to change bound ...
clelab 2405 Membership of a class vari...
clabel 2406 Membership of a class abst...
sbab 2407 The right-hand side of the...
nfcjust 2409 Justification theorem for ...
nfci 2411 Deduce that a class ` A ` ...
nfcii 2412 Deduce that a class ` A ` ...
nfcr 2413 Consequence of the not-fre...
nfcrii 2414 Consequence of the not-fre...
nfcri 2415 Consequence of the not-fre...
nfcd 2416 Deduce that a class ` A ` ...
nfceqi 2417 Equality theorem for class...
nfcxfr 2418 A utility lemma to transfe...
nfcxfrd 2419 A utility lemma to transfe...
nfceqdf 2420 An equality theorem for ef...
nfcv 2421 If ` x ` is disjoint from ...
nfcvd 2422 If ` x ` is disjoint from ...
nfab1 2423 Bound-variable hypothesis ...
nfnfc1 2424 ` x ` is bound in ` F/_ x ...
nfab 2425 Bound-variable hypothesis ...
nfaba1 2426 Bound-variable hypothesis ...
nfnfc 2427 Hypothesis builder for ` F...
nfeq 2428 Hypothesis builder for equ...
nfel 2429 Hypothesis builder for ele...
nfeq1 2430 Hypothesis builder for equ...
nfel1 2431 Hypothesis builder for ele...
nfeq2 2432 Hypothesis builder for equ...
nfel2 2433 Hypothesis builder for ele...
nfcrd 2434 Consequence of the not-fre...
nfeqd 2435 Hypothesis builder for equ...
nfeld 2436 Hypothesis builder for ele...
drnfc1 2437 Formula-building lemma for...
drnfc2 2438 Formula-building lemma for...
nfabd2 2439 Bound-variable hypothesis ...
nfabd 2440 Bound-variable hypothesis ...
dvelimdc 2441 Deduction form of ~ dvelim...
dvelimc 2442 Version of ~ dvelim for cl...
nfcvf 2443 If ` x ` and ` y ` are dis...
nfcvf2 2444 If ` x ` and ` y ` are dis...
cleqf 2445 Establish equality between...
abid2f 2446 A simplification of class ...
sbabel 2447 Theorem to move a substitu...
nne 2452 Negation of inequality. (...
neirr 2453 No class is unequal to its...
exmidne 2454 Excluded middle with equal...
nonconne 2455 Law of noncontradiction wi...
neeq1 2456 Equality theorem for inequ...
neeq2 2457 Equality theorem for inequ...
neeq1i 2458 Inference for inequality. ...
neeq2i 2459 Inference for inequality. ...
neeq12i 2460 Inference for inequality. ...
neeq1d 2461 Deduction for inequality. ...
neeq2d 2462 Deduction for inequality. ...
neeq12d 2463 Deduction for inequality. ...
neneqd 2464 Deduction eliminating ineq...
eqnetri 2465 Substitution of equal clas...
eqnetrd 2466 Substitution of equal clas...
eqnetrri 2467 Substitution of equal clas...
eqnetrrd 2468 Substitution of equal clas...
neeqtri 2469 Substitution of equal clas...
neeqtrd 2470 Substitution of equal clas...
neeqtrri 2471 Substitution of equal clas...
neeqtrrd 2472 Substitution of equal clas...
syl5eqner 2473 B chained equality inferen...
3netr3d 2474 Substitution of equality i...
3netr4d 2475 Substitution of equality i...
3netr3g 2476 Substitution of equality i...
3netr4g 2477 Substitution of equality i...
necon3abii 2478 Deduction from equality to...
necon3bbii 2479 Deduction from equality to...
necon3bii 2480 Inference from equality to...
necon3abid 2481 Deduction from equality to...
necon3bbid 2482 Deduction from equality to...
necon3bid 2483 Deduction from equality to...
necon3ad 2484 Contrapositive law deducti...
necon3bd 2485 Contrapositive law deducti...
necon3d 2486 Contrapositive law deducti...
necon3i 2487 Contrapositive inference f...
necon3ai 2488 Contrapositive inference f...
necon3bi 2489 Contrapositive inference f...
necon1ai 2490 Contrapositive inference f...
necon1bi 2491 Contrapositive inference f...
necon1i 2492 Contrapositive inference f...
necon2ai 2493 Contrapositive inference f...
necon2bi 2494 Contrapositive inference f...
necon2i 2495 Contrapositive inference f...
necon2ad 2496 Contrapositive inference f...
necon2bd 2497 Contrapositive inference f...
necon2d 2498 Contrapositive inference f...
necon1abii 2499 Contrapositive inference f...
necon1bbii 2500 Contrapositive inference f...
necon1abid 2501 Contrapositive deduction f...
necon1bbid 2502 Contrapositive inference f...
necon2abii 2503 Contrapositive inference f...
necon2bbii 2504 Contrapositive inference f...
necon2abid 2505 Contrapositive deduction f...
necon2bbid 2506 Contrapositive deduction f...
necon4ai 2507 Contrapositive inference f...
necon4i 2508 Contrapositive inference f...
necon4ad 2509 Contrapositive inference f...
necon4bd 2510 Contrapositive inference f...
necon4d 2511 Contrapositive inference f...
necon4abid 2512 Contrapositive law deducti...
necon4bbid 2513 Contrapositive law deducti...
necon4bid 2514 Contrapositive law deducti...
necon1ad 2515 Contrapositive deduction f...
necon1bd 2516 Contrapositive deduction f...
necon1d 2517 Contrapositive law deducti...
neneqad 2518 If it is not the case that...
nebi 2519 Contraposition law for ine...
pm13.18 2520 Theorem *13.18 in [Whitehe...
pm13.181 2521 Theorem *13.181 in [Whiteh...
pm2.21ddne 2522 A contradiction implies an...
pm2.61ne 2523 Deduction eliminating an i...
pm2.61ine 2524 Inference eliminating an i...
pm2.61dne 2525 Deduction eliminating an i...
pm2.61dane 2526 Deduction eliminating an i...
pm2.61da2ne 2527 Deduction eliminating two ...
pm2.61da3ne 2528 Deduction eliminating thre...
necom 2529 Commutation of inequality....
necomi 2530 Inference from commutative...
necomd 2531 Deduction from commutative...
neor 2532 Logical OR with an equalit...
neanior 2533 A De Morgan's law for ineq...
ne3anior 2534 A De Morgan's law for ineq...
neorian 2535 A De Morgan's law for ineq...
nemtbir 2536 An inference from an inequ...
nelne1 2537 Two classes are different ...
nelne2 2538 Two classes are different ...
neleq1 2539 Equality theorem for negat...
neleq2 2540 Equality theorem for negat...
nfne 2541 Bound-variable hypothesis ...
nfnel 2542 Bound-variable hypothesis ...
nfned 2543 Bound-variable hypothesis ...
nfneld 2544 Bound-variable hypothesis ...
ralnex 2555 Relationship between restr...
rexnal 2556 Relationship between restr...
dfral2 2557 Relationship between restr...
dfrex2 2558 Relationship between restr...
ralbida 2559 Formula-building rule for ...
rexbida 2560 Formula-building rule for ...
ralbidva 2561 Formula-building rule for ...
rexbidva 2562 Formula-building rule for ...
ralbid 2563 Formula-building rule for ...
rexbid 2564 Formula-building rule for ...
ralbidv 2565 Formula-building rule for ...
rexbidv 2566 Formula-building rule for ...
ralbidv2 2567 Formula-building rule for ...
rexbidv2 2568 Formula-building rule for ...
ralbii 2569 Inference adding restricte...
rexbii 2570 Inference adding restricte...
2ralbii 2571 Inference adding two restr...
2rexbii 2572 Inference adding two restr...
ralbii2 2573 Inference adding different...
rexbii2 2574 Inference adding different...
raleqbii 2575 Equality deduction for res...
rexeqbii 2576 Equality deduction for res...
ralbiia 2577 Inference adding restricte...
rexbiia 2578 Inference adding restricte...
2rexbiia 2579 Inference adding two restr...
r2alf 2580 Double restricted universa...
r2exf 2581 Double restricted existent...
r2al 2582 Double restricted universa...
r2ex 2583 Double restricted existent...
2ralbida 2584 Formula-building rule for ...
2ralbidva 2585 Formula-building rule for ...
2rexbidva 2586 Formula-building rule for ...
2ralbidv 2587 Formula-building rule for ...
2rexbidv 2588 Formula-building rule for ...
rexralbidv 2589 Formula-building rule for ...
ralinexa 2590 A transformation of restri...
rexanali 2591 A transformation of restri...
risset 2592 Two ways to say " ` A ` be...
hbral 2593 Bound-variable hypothesis ...
hbra1 2594 ` x ` is not free in ` A. ...
nfra1 2595 ` x ` is not free in ` A. ...
nfrald 2596 Deduction version of ~ nfr...
nfrexd 2597 Deduction version of ~ nfr...
nfral 2598 Bound-variable hypothesis ...
nfra2 2599 Similar to Lemma 24 of [Mo...
nfrex 2600 Bound-variable hypothesis ...
nfre1 2601 ` x ` is not free in ` E. ...
r3al 2602 Triple restricted universa...
alral 2603 Universal quantification i...
rexex 2604 Restricted existence impli...
rsp 2605 Restricted specialization....
rspe 2606 Restricted specialization....
rsp2 2607 Restricted specialization....
rsp2e 2608 Restricted specialization....
rspec 2609 Specialization rule for re...
rgen 2610 Generalization rule for re...
rgen2a 2611 Generalization rule for re...
rgenw 2612 Generalization rule for re...
rgen2w 2613 Generalization rule for re...
mprg 2614 Modus ponens combined with...
mprgbir 2615 Modus ponens on biconditio...
ralim 2616 Distribution of restricted...
ralimi2 2617 Inference quantifying both...
ralimia 2618 Inference quantifying both...
ralimiaa 2619 Inference quantifying both...
ralimi 2620 Inference quantifying both...
ral2imi 2621 Inference quantifying ante...
ralimdaa 2622 Deduction quantifying both...
ralimdva 2623 Deduction quantifying both...
ralimdv 2624 Deduction quantifying both...
ralimdv2 2625 Inference quantifying both...
ralrimi 2626 Inference from Theorem 19....
ralrimiv 2627 Inference from Theorem 19....
ralrimiva 2628 Inference from Theorem 19....
ralrimivw 2629 Inference from Theorem 19....
r19.21t 2630 Theorem 19.21 of [Margaris...
r19.21 2631 Theorem 19.21 of [Margaris...
r19.21v 2632 Theorem 19.21 of [Margaris...
ralrimd 2633 Inference from Theorem 19....
ralrimdv 2634 Inference from Theorem 19....
ralrimdva 2635 Inference from Theorem 19....
ralrimivv 2636 Inference from Theorem 19....
ralrimivva 2637 Inference from Theorem 19....
ralrimivvva 2638 Inference from Theorem 19....
ralrimdvv 2639 Inference from Theorem 19....
ralrimdvva 2640 Inference from Theorem 19....
rgen2 2641 Generalization rule for re...
rgen3 2642 Generalization rule for re...
r19.21bi 2643 Inference from Theorem 19....
rspec2 2644 Specialization rule for re...
rspec3 2645 Specialization rule for re...
r19.21be 2646 Inference from Theorem 19....
nrex 2647 Inference adding restricte...
nrexdv 2648 Deduction adding restricte...
rexim 2649 Theorem 19.22 of [Margaris...
reximia 2650 Inference quantifying both...
reximi2 2651 Inference quantifying both...
reximi 2652 Inference quantifying both...
reximdai 2653 Deduction from Theorem 19....
reximdv2 2654 Deduction quantifying both...
reximdvai 2655 Deduction quantifying both...
reximdv 2656 Deduction from Theorem 19....
reximdva 2657 Deduction quantifying both...
r19.12 2658 Theorem 19.12 of [Margaris...
r19.23t 2659 Closed theorem form of ~ r...
r19.23 2660 Theorem 19.23 of [Margaris...
r19.23v 2661 Theorem 19.23 of [Margaris...
rexlimi 2662 Inference from Theorem 19....
rexlimiv 2663 Inference from Theorem 19....
rexlimiva 2664 Inference from Theorem 19....
rexlimivw 2665 Weaker version of ~ rexlim...
rexlimd 2666 Deduction from Theorem 19....
rexlimd2 2667 Version of ~ rexlimd with ...
rexlimdv 2668 Inference from Theorem 19....
rexlimdva 2669 Inference from Theorem 19....
rexlimdvaa 2670 Inference from Theorem 19....
rexlimdv3a 2671 Inference from Theorem 19....
rexlimdvw 2672 Inference from Theorem 19....
rexlimddv 2673 Restricted existential eli...
rexlimivv 2674 Inference from Theorem 19....
rexlimdvv 2675 Inference from Theorem 19....
rexlimdvva 2676 Inference from Theorem 19....
r19.26 2677 Theorem 19.26 of [Margaris...
r19.26-2 2678 Theorem 19.26 of [Margaris...
r19.26-3 2679 Theorem 19.26 of [Margaris...
r19.26m 2680 Theorem 19.26 of [Margaris...
ralbi 2681 Distribute a restricted un...
ralbiim 2682 Split a biconditional and ...
r19.27av 2683 Restricted version of one ...
r19.28av 2684 Restricted version of one ...
r19.29 2685 Theorem 19.29 of [Margaris...
r19.29r 2686 Variation of Theorem 19.29...
r19.30 2687 Theorem 19.30 of [Margaris...
r19.32v 2688 Theorem 19.32 of [Margaris...
r19.35 2689 Restricted quantifier vers...
r19.36av 2690 One direction of a restric...
r19.37 2691 Restricted version of one ...
r19.37av 2692 Restricted version of one ...
r19.40 2693 Restricted quantifier vers...
r19.41 2694 Restricted quantifier vers...
r19.41v 2695 Restricted quantifier vers...
r19.42v 2696 Restricted version of Theo...
r19.43 2697 Restricted version of Theo...
r19.44av 2698 One direction of a restric...
r19.45av 2699 Restricted version of one ...
ralcomf 2700 Commutation of restricted ...
rexcomf 2701 Commutation of restricted ...
ralcom 2702 Commutation of restricted ...
rexcom 2703 Commutation of restricted ...
rexcom13 2704 Swap 1st and 3rd restricte...
rexrot4 2705 Rotate existential restric...
ralcom2 2706 Commutation of restricted ...
ralcom3 2707 A commutative law for rest...
reean 2708 Rearrange existential quan...
reeanv 2709 Rearrange existential quan...
3reeanv 2710 Rearrange three existentia...
2ralor 2711 Distribute quantification ...
nfreu1 2712 ` x ` is not free in ` E! ...
nfrmo1 2713 ` x ` is not free in ` E* ...
nfreud 2714 Deduction version of ~ nfr...
nfrmod 2715 Deduction version of ~ nfr...
nfreu 2716 Bound-variable hypothesis ...
nfrmo 2717 Bound-variable hypothesis ...
rabid 2718 An "identity" law of concr...
rabid2 2719 An "identity" law for rest...
rabbi 2720 Equivalent wff's correspon...
rabswap 2721 Swap with a membership rel...
nfrab1 2722 The abstraction variable i...
nfrab 2723 A variable not free in a w...
reubida 2724 Formula-building rule for ...
reubidva 2725 Formula-building rule for ...
reubidv 2726 Formula-building rule for ...
reubiia 2727 Formula-building rule for ...
reubii 2728 Formula-building rule for ...
rmobida 2729 Formula-building rule for ...
rmobidva 2730 Formula-building rule for ...
rmobidv 2731 Formula-building rule for ...
rmobiia 2732 Formula-building rule for ...
rmobii 2733 Formula-building rule for ...
raleqf 2734 Equality theorem for restr...
rexeqf 2735 Equality theorem for restr...
reueq1f 2736 Equality theorem for restr...
rmoeq1f 2737 Equality theorem for restr...
raleq 2738 Equality theorem for restr...
rexeq 2739 Equality theorem for restr...
reueq1 2740 Equality theorem for restr...
rmoeq1 2741 Equality theorem for restr...
raleqi 2742 Equality inference for res...
rexeqi 2743 Equality inference for res...
raleqdv 2744 Equality deduction for res...
rexeqdv 2745 Equality deduction for res...
raleqbi1dv 2746 Equality deduction for res...
rexeqbi1dv 2747 Equality deduction for res...
reueqd 2748 Equality deduction for res...
rmoeqd 2749 Equality deduction for res...
raleqbidv 2750 Equality deduction for res...
rexeqbidv 2751 Equality deduction for res...
raleqbidva 2752 Equality deduction for res...
rexeqbidva 2753 Equality deduction for res...
mormo 2754 Unrestricted "at most one"...
reu5 2755 Restricted uniqueness in t...
reurex 2756 Restricted unique existenc...
reurmo 2757 Restricted existential uni...
rmo5 2758 Restricted "at most one" i...
nrexrmo 2759 Nonexistence implies restr...
cbvralf 2760 Rule used to change bound ...
cbvrexf 2761 Rule used to change bound ...
cbvral 2762 Rule used to change bound ...
cbvrex 2763 Rule used to change bound ...
cbvreu 2764 Change the bound variable ...
cbvrmo 2765 Change the bound variable ...
cbvralv 2766 Change the bound variable ...
cbvrexv 2767 Change the bound variable ...
cbvreuv 2768 Change the bound variable ...
cbvrmov 2769 Change the bound variable ...
cbvraldva2 2770 Rule used to change the bo...
cbvrexdva2 2771 Rule used to change the bo...
cbvraldva 2772 Rule used to change the bo...
cbvrexdva 2773 Rule used to change the bo...
cbvral2v 2774 Change bound variables of ...
cbvrex2v 2775 Change bound variables of ...
cbvral3v 2776 Change bound variables of ...
cbvralsv 2777 Change bound variable by u...
cbvrexsv 2778 Change bound variable by u...
sbralie 2779 Implicit to explicit subst...
rabbiia 2780 Equivalent wff's yield equ...
rabbidva 2781 Equivalent wff's yield equ...
rabbidv 2782 Equivalent wff's yield equ...
rabeqf 2783 Equality theorem for restr...
rabeq 2784 Equality theorem for restr...
rabeqbidv 2785 Equality of restricted cla...
rabeqbidva 2786 Equality of restricted cla...
rabeq2i 2787 Inference rule from equali...
cbvrab 2788 Rule to change the bound v...
cbvrabv 2789 Rule to change the bound v...
vjust 2791 Soundness justification th...
vex 2793 All set variables are sets...
isset 2794 Two ways to say " ` A ` is...
issetf 2795 A version of isset that do...
isseti 2796 A way to say " ` A ` is a ...
issetri 2797 A way to say " ` A ` is a ...
elex 2798 If a class is a member of ...
elexi 2799 If a class is a member of ...
elisset 2800 An element of a class exis...
elex22 2801 If two classes each contai...
elex2 2802 If a class contains anothe...
ralv 2803 A universal quantifier res...
rexv 2804 An existential quantifier ...
reuv 2805 A uniqueness quantifier re...
rmov 2806 A uniqueness quantifier re...
rabab 2807 A class abstraction restri...
ralcom4 2808 Commutation of restricted ...
rexcom4 2809 Commutation of restricted ...
rexcom4a 2810 Specialized existential co...
rexcom4b 2811 Specialized existential co...
ceqsalt 2812 Closed theorem version of ...
ceqsralt 2813 Restricted quantifier vers...
ceqsalg 2814 A representation of explic...
ceqsal 2815 A representation of explic...
ceqsalv 2816 A representation of explic...
ceqsralv 2817 Restricted quantifier vers...
gencl 2818 Implicit substitution for ...
2gencl 2819 Implicit substitution for ...
3gencl 2820 Implicit substitution for ...
cgsexg 2821 Implicit substitution infe...
cgsex2g 2822 Implicit substitution infe...
cgsex4g 2823 An implicit substitution i...
ceqsex 2824 Elimination of an existent...
ceqsexv 2825 Elimination of an existent...
ceqsex2 2826 Elimination of two existen...
ceqsex2v 2827 Elimination of two existen...
ceqsex3v 2828 Elimination of three exist...
ceqsex4v 2829 Elimination of four existe...
ceqsex6v 2830 Elimination of six existen...
ceqsex8v 2831 Elimination of eight exist...
gencbvex 2832 Change of bound variable u...
gencbvex2 2833 Restatement of ~ gencbvex ...
gencbval 2834 Change of bound variable u...
sbhypf 2835 Introduce an explicit subs...
vtoclgft 2836 Closed theorem form of ~ v...
vtocldf 2837 Implicit substitution of a...
vtocld 2838 Implicit substitution of a...
vtoclf 2839 Implicit substitution of a...
vtocl 2840 Implicit substitution of a...
vtocl2 2841 Implicit substitution of c...
vtocl3 2842 Implicit substitution of c...
vtoclb 2843 Implicit substitution of a...
vtoclgf 2844 Implicit substitution of a...
vtoclg 2845 Implicit substitution of a...
vtoclbg 2846 Implicit substitution of a...
vtocl2gf 2847 Implicit substitution of a...
vtocl3gf 2848 Implicit substitution of a...
vtocl2g 2849 Implicit substitution of 2...
vtoclgaf 2850 Implicit substitution of a...
vtoclga 2851 Implicit substitution of a...
vtocl2gaf 2852 Implicit substitution of 2...
vtocl2ga 2853 Implicit substitution of 2...
vtocl3gaf 2854 Implicit substitution of 3...
vtocl3ga 2855 Implicit substitution of 3...
vtocleg 2856 Implicit substitution of a...
vtoclegft 2857 Implicit substitution of a...
vtoclef 2858 Implicit substitution of a...
vtocle 2859 Implicit substitution of a...
vtoclri 2860 Implicit substitution of a...
spcimgft 2861 A closed version of ~ spci...
spcgft 2862 A closed version of ~ spcg...
spcimgf 2863 Rule of specialization, us...
spcimegf 2864 Existential specialization...
spcgf 2865 Rule of specialization, us...
spcegf 2866 Existential specialization...
spcimdv 2867 Restricted specialization,...
spcdv 2868 Rule of specialization, us...
spcimedv 2869 Restricted existential spe...
spcgv 2870 Rule of specialization, us...
spcegv 2871 Existential specialization...
spc2egv 2872 Existential specialization...
spc2gv 2873 Specialization with 2 quan...
spc3egv 2874 Existential specialization...
spc3gv 2875 Specialization with 3 quan...
spcv 2876 Rule of specialization, us...
spcev 2877 Existential specialization...
spc2ev 2878 Existential specialization...
rspct 2879 A closed version of ~ rspc...
rspc 2880 Restricted specialization,...
rspce 2881 Restricted existential spe...
rspcv 2882 Restricted specialization,...
rspccv 2883 Restricted specialization,...
rspcva 2884 Restricted specialization,...
rspccva 2885 Restricted specialization,...
rspcev 2886 Restricted existential spe...
rspcimdv 2887 Restricted specialization,...
rspcimedv 2888 Restricted existential spe...
rspcdv 2889 Restricted specialization,...
rspcedv 2890 Restricted existential spe...
rspc2 2891 2-variable restricted spec...
rspc2v 2892 2-variable restricted spec...
rspc2va 2893 2-variable restricted spec...
rspc2ev 2894 2-variable restricted exis...
rspc3v 2895 3-variable restricted spec...
rspc3ev 2896 3-variable restricted exis...
eqvinc 2897 A variable introduction la...
eqvincf 2898 A variable introduction la...
alexeq 2899 Two ways to express substi...
ceqex 2900 Equality implies equivalen...
ceqsexg 2901 A representation of explic...
ceqsexgv 2902 Elimination of an existent...
ceqsrexv 2903 Elimination of a restricte...
ceqsrexbv 2904 Elimination of a restricte...
ceqsrex2v 2905 Elimination of a restricte...
clel2 2906 An alternate definition of...
clel3g 2907 An alternate definition of...
clel3 2908 An alternate definition of...
clel4 2909 An alternate definition of...
pm13.183 2910 Compare theorem *13.183 in...
rr19.3v 2911 Restricted quantifier vers...
rr19.28v 2912 Restricted quantifier vers...
elabgt 2913 Membership in a class abst...
elabgf 2914 Membership in a class abst...
elabf 2915 Membership in a class abst...
elab 2916 Membership in a class abst...
elabg 2917 Membership in a class abst...
elab2g 2918 Membership in a class abst...
elab2 2919 Membership in a class abst...
elab4g 2920 Membership in a class abst...
elab3gf 2921 Membership in a class abst...
elab3g 2922 Membership in a class abst...
elab3 2923 Membership in a class abst...
elrabf 2924 Membership in a restricted...
elrab 2925 Membership in a restricted...
elrab3 2926 Membership in a restricted...
elrab2 2927 Membership in a class abst...
ralab 2928 Universal quantification o...
ralrab 2929 Universal quantification o...
rexab 2930 Existential quantification...
rexrab 2931 Existential quantification...
ralab2 2932 Universal quantification o...
ralrab2 2933 Universal quantification o...
rexab2 2934 Existential quantification...
rexrab2 2935 Existential quantification...
abidnf 2936 Identity used to create cl...
dedhb 2937 A deduction theorem for co...
eqeu 2938 A condition which implies ...
eueq 2939 Equality has existential u...
eueq1 2940 Equality has existential u...
eueq2 2941 Equality has existential u...
eueq3 2942 Equality has existential u...
moeq 2943 There is at most one set e...
moeq3 2944 "At most one" property of ...
mosub 2945 "At most one" remains true...
mo2icl 2946 Theorem for inferring "at ...
mob2 2947 Consequence of "at most on...
moi2 2948 Consequence of "at most on...
mob 2949 Equality implied by "at mo...
moi 2950 Equality implied by "at mo...
morex 2951 Derive membership from uni...
euxfr2 2952 Transfer existential uniqu...
euxfr 2953 Transfer existential uniqu...
euind 2954 Existential uniqueness via...
reu2 2955 A way to express restricte...
reu6 2956 A way to express restricte...
reu3 2957 A way to express restricte...
reu6i 2958 A condition which implies ...
eqreu 2959 A condition which implies ...
rmo4 2960 Restricted "at most one" u...
reu4 2961 Restricted uniqueness usin...
reu7 2962 Restricted uniqueness usin...
reu8 2963 Restricted uniqueness usin...
reueq 2964 Equality has existential u...
rmoan 2965 Restricted "at most one" s...
rmoim 2966 Restricted "at most one" i...
rmoimia 2967 Restricted "at most one" i...
rmoimi2 2968 Restricted "at most one" i...
2reuswap 2969 A condition allowing swap ...
reuind 2970 Existential uniqueness via...
2rmorex 2971 Double restricted quantifi...
2reu5lem1 2972 Lemma for ~ 2reu5 . Note ...
2reu5lem2 2973 Lemma for ~ 2reu5 . (Cont...
2reu5lem3 2974 Lemma for ~ 2reu5 . This ...
2reu5 2975 Double restricted existent...
cdeqi 2978 Deduce conditional equalit...
cdeqri 2979 Property of conditional eq...
cdeqth 2980 Deduce conditional equalit...
cdeqnot 2981 Distribute conditional equ...
cdeqal 2982 Distribute conditional equ...
cdeqab 2983 Distribute conditional equ...
cdeqal1 2984 Distribute conditional equ...
cdeqab1 2985 Distribute conditional equ...
cdeqim 2986 Distribute conditional equ...
cdeqcv 2987 Conditional equality for s...
cdeqeq 2988 Distribute conditional equ...
cdeqel 2989 Distribute conditional equ...
nfcdeq 2990 If we have a conditional e...
nfccdeq 2991 Variation of ~ nfcdeq for ...
ru 2992 Russell's Paradox. Propos...
dfsbcq 2995 This theorem, which is sim...
dfsbcq2 2996 This theorem, which is sim...
sbsbc 2997 Show that ~ df-sb and ~ df...
sbceq1d 2998 Equality theorem for class...
sbceq1dd 2999 Equality theorem for class...
sbc8g 3000 This is the closest we can...
sbc2or 3001 The disjunction of two equ...
sbcex 3002 By our definition of prope...
sbceq1a 3003 Equality theorem for class...
sbceq2a 3004 Equality theorem for class...
spsbc 3005 Specialization: if a formu...
spsbcd 3006 Specialization: if a formu...
sbcth 3007 A substitution into a theo...
sbcthdv 3008 Deduction version of ~ sbc...
sbcid 3009 An identity theorem for su...
nfsbc1d 3010 Deduction version of ~ nfs...
nfsbc1 3011 Bound-variable hypothesis ...
nfsbc1v 3012 Bound-variable hypothesis ...
nfsbcd 3013 Deduction version of ~ nfs...
nfsbc 3014 Bound-variable hypothesis ...
sbcco 3015 A composition law for clas...
sbcco2 3016 A composition law for clas...
sbc5 3017 An equivalence for class s...
sbc6g 3018 An equivalence for class s...
sbc6 3019 An equivalence for class s...
sbc7 3020 An equivalence for class s...
cbvsbc 3021 Change bound variables in ...
cbvsbcv 3022 Change the bound variable ...
sbciegft 3023 Conversion of implicit sub...
sbciegf 3024 Conversion of implicit sub...
sbcieg 3025 Conversion of implicit sub...
sbcie2g 3026 Conversion of implicit sub...
sbcie 3027 Conversion of implicit sub...
sbciedf 3028 Conversion of implicit sub...
sbcied 3029 Conversion of implicit sub...
sbcied2 3030 Conversion of implicit sub...
elrabsf 3031 Membership in a restricted...
eqsbc3 3032 Substitution applied to an...
sbcng 3033 Move negation in and out o...
sbcimg 3034 Distribution of class subs...
sbcan 3035 Distribution of class subs...
sbcang 3036 Distribution of class subs...
sbcor 3037 Distribution of class subs...
sbcorg 3038 Distribution of class subs...
sbcbig 3039 Distribution of class subs...
sbcal 3040 Move universal quantifier ...
sbcalg 3041 Move universal quantifier ...
sbcex2 3042 Move existential quantifie...
sbcexg 3043 Move existential quantifie...
sbceqal 3044 Set theory version of ~ sb...
sbeqalb 3045 Theorem *14.121 in [Whiteh...
sbcbid 3046 Formula-building deduction...
sbcbidv 3047 Formula-building deduction...
sbcbii 3048 Formula-building inference...
sbcbiiOLD 3049 Formula-building inference...
eqsbc3r 3050 ~ eqsbc3 with set variable...
sbc3ang 3051 Distribution of class subs...
sbcel1gv 3052 Class substitution into a ...
sbcel2gv 3053 Class substitution into a ...
sbcimdv 3054 Substitution analog of The...
sbctt 3055 Substitution for a variabl...
sbcgf 3056 Substitution for a variabl...
sbc19.21g 3057 Substitution for a variabl...
sbcg 3058 Substitution for a variabl...
sbc2iegf 3059 Conversion of implicit sub...
sbc2ie 3060 Conversion of implicit sub...
sbc2iedv 3061 Conversion of implicit sub...
sbc3ie 3062 Conversion of implicit sub...
sbccomlem 3063 Lemma for ~ sbccom . (Con...
sbccom 3064 Commutative law for double...
sbcralt 3065 Interchange class substitu...
sbcrext 3066 Interchange class substitu...
sbcralg 3067 Interchange class substitu...
sbcrexg 3068 Interchange class substitu...
sbcreug 3069 Interchange class substitu...
sbcabel 3070 Interchange class substitu...
rspsbc 3071 Restricted quantifier vers...
rspsbca 3072 Restricted quantifier vers...
rspesbca 3073 Existence form of ~ rspsbc...
spesbc 3074 Existence form of ~ spsbc ...
spesbcd 3075 form of ~ spsbc . (Contri...
sbcth2 3076 A substitution into a theo...
ra5 3077 Restricted quantifier vers...
rmo2 3078 Alternate definition of re...
rmo2i 3079 Condition implying restric...
rmo3 3080 Restricted "at most one" u...
rmob 3081 Consequence of "at most on...
rmoi 3082 Consequence of "at most on...
csb2 3085 Alternate expression for t...
csbeq1 3086 Analog of ~ dfsbcq for pro...
cbvcsb 3087 Change bound variables in ...
cbvcsbv 3088 Change the bound variable ...
csbeq1d 3089 Equality deduction for pro...
csbid 3090 Analog of ~ sbid for prope...
csbeq1a 3091 Equality theorem for prope...
csbco 3092 Composition law for chaine...
csbexg 3093 The existence of proper su...
csbex 3094 The existence of proper su...
csbtt 3095 Substitution doesn't affec...
csbconstgf 3096 Substitution doesn't affec...
csbconstg 3097 Substitution doesn't affec...
sbcel12g 3098 Distribute proper substitu...
sbceqg 3099 Distribute proper substitu...
sbcnel12g 3100 Distribute proper substitu...
sbcne12g 3101 Distribute proper substitu...
sbcel1g 3102 Move proper substitution i...
sbceq1g 3103 Move proper substitution t...
sbcel2g 3104 Move proper substitution i...
sbceq2g 3105 Move proper substitution t...
csbcomg 3106 Commutative law for double...
csbeq2d 3107 Formula-building deduction...
csbeq2dv 3108 Formula-building deduction...
csbeq2i 3109 Formula-building inference...
csbvarg 3110 The proper substitution of...
sbccsbg 3111 Substitution into a wff ex...
sbccsb2g 3112 Substitution into a wff ex...
nfcsb1d 3113 Bound-variable hypothesis ...
nfcsb1 3114 Bound-variable hypothesis ...
nfcsb1v 3115 Bound-variable hypothesis ...
nfcsbd 3116 Deduction version of ~ nfc...
nfcsb 3117 Bound-variable hypothesis ...
csbhypf 3118 Introduce an explicit subs...
csbiebt 3119 Conversion of implicit sub...
csbiedf 3120 Conversion of implicit sub...
csbieb 3121 Bidirectional conversion b...
csbiebg 3122 Bidirectional conversion b...
csbiegf 3123 Conversion of implicit sub...
csbief 3124 Conversion of implicit sub...
csbied 3125 Conversion of implicit sub...
csbied2 3126 Conversion of implicit sub...
csbie2t 3127 Conversion of implicit sub...
csbie2 3128 Conversion of implicit sub...
csbie2g 3129 Conversion of implicit sub...
sbcnestgf 3130 Nest the composition of tw...
csbnestgf 3131 Nest the composition of tw...
sbcnestg 3132 Nest the composition of tw...
csbnestg 3133 Nest the composition of tw...
csbnestgOLD 3134 Nest the composition of tw...
csbnest1g 3135 Nest the composition of tw...
csbnest1gOLD 3136 Nest the composition of tw...
csbidmg 3137 Idempotent law for class s...
sbcco3g 3138 Composition of two substit...
sbcco3gOLD 3139 Composition of two substit...
csbco3g 3140 Composition of two class s...
csbco3gOLD 3141 Composition of two class s...
rspcsbela 3142 Special case related to ~ ...
sbnfc2 3143 Two ways of expressing " `...
csbabg 3144 Move substitution into a c...
cbvralcsf 3145 A more general version of ...
cbvrexcsf 3146 A more general version of ...
cbvreucsf 3147 A more general version of ...
cbvrabcsf 3148 A more general version of ...
cbvralv2 3149 Rule used to change the bo...
cbvrexv2 3150 Rule used to change the bo...
difjust 3156 Soundness justification th...
unjust 3158 Soundness justification th...
injust 3160 Soundness justification th...
dfin5 3162 Alternate definition for t...
dfdif2 3163 Alternate definition of cl...
eldif 3164 Expansion of membership in...
eldifd 3165 If a class is in one class...
eldifad 3166 If a class is in the diffe...
eldifbd 3167 If a class is in the diffe...
dfss 3169 Variant of subclass defini...
dfss2 3171 Alternate definition of th...
dfss3 3172 Alternate definition of su...
dfss2f 3173 Equivalence for subclass r...
dfss3f 3174 Equivalence for subclass r...
nfss 3175 If ` x ` is not free in ` ...
ssel 3176 Membership relationships f...
ssel2 3177 Membership relationships f...
sseli 3178 Membership inference from ...
sselii 3179 Membership inference from ...
sseldi 3180 Membership inference from ...
sseld 3181 Membership deduction from ...
sselda 3182 Membership deduction from ...
sseldd 3183 Membership inference from ...
ssneld 3184 If a class is not in anoth...
ssneldd 3185 If an element is not in a ...
ssriv 3186 Inference rule based on su...
ssrdv 3187 Deduction rule based on su...
sstr2 3188 Transitivity of subclasses...
sstr 3189 Transitivity of subclasses...
sstri 3190 Subclass transitivity infe...
sstrd 3191 Subclass transitivity dedu...
syl5ss 3192 Subclass transitivity dedu...
syl6ss 3193 Subclass transitivity dedu...
sylan9ss 3194 A subclass transitivity de...
sylan9ssr 3195 A subclass transitivity de...
eqss 3196 The subclass relationship ...
eqssi 3197 Infer equality from two su...
eqssd 3198 Equality deduction from tw...
ssid 3199 Any class is a subclass of...
ssv 3200 Any class is a subclass of...
sseq1 3201 Equality theorem for subcl...
sseq2 3202 Equality theorem for the s...
sseq12 3203 Equality theorem for the s...
sseq1i 3204 An equality inference for ...
sseq2i 3205 An equality inference for ...
sseq12i 3206 An equality inference for ...
sseq1d 3207 An equality deduction for ...
sseq2d 3208 An equality deduction for ...
sseq12d 3209 An equality deduction for ...
eqsstri 3210 Substitution of equality i...
eqsstr3i 3211 Substitution of equality i...
sseqtri 3212 Substitution of equality i...
sseqtr4i 3213 Substitution of equality i...
eqsstrd 3214 Substitution of equality i...
eqsstr3d 3215 Substitution of equality i...
sseqtrd 3216 Substitution of equality i...
sseqtr4d 3217 Substitution of equality i...
3sstr3i 3218 Substitution of equality i...
3sstr4i 3219 Substitution of equality i...
3sstr3g 3220 Substitution of equality i...
3sstr4g 3221 Substitution of equality i...
3sstr3d 3222 Substitution of equality i...
3sstr4d 3223 Substitution of equality i...
syl5eqss 3224 B chained subclass and equ...
syl5eqssr 3225 B chained subclass and equ...
syl6sseq 3226 A chained subclass and equ...
syl6sseqr 3227 A chained subclass and equ...
syl5sseq 3228 Subclass transitivity dedu...
syl5sseqr 3229 Subclass transitivity dedu...
syl6eqss 3230 A chained subclass and equ...
syl6eqssr 3231 A chained subclass and equ...
eqimss 3232 Equality implies the subcl...
eqimss2 3233 Equality implies the subcl...
eqimssi 3234 Infer subclass relationshi...
eqimss2i 3235 Infer subclass relationshi...
nssne1 3236 Two classes are different ...
nssne2 3237 Two classes are different ...
nss 3238 Negation of subclass relat...
ssralv 3239 Quantification restricted ...
ssrexv 3240 Existential quantification...
ralss 3241 Restricted universal quant...
rexss 3242 Restricted existential qua...
ss2ab 3243 Class abstractions in a su...
abss 3244 Class abstraction in a sub...
ssab 3245 Subclass of a class abstra...
ssabral 3246 The relation for a subclas...
ss2abi 3247 Inference of abstraction s...
ss2abdv 3248 Deduction of abstraction s...
abssdv 3249 Deduction of abstraction s...
abssi 3250 Inference of abstraction s...
ss2rab 3251 Restricted abstraction cla...
rabss 3252 Restricted class abstracti...
ssrab 3253 Subclass of a restricted c...
ssrabdv 3254 Subclass of a restricted c...
rabssdv 3255 Subclass of a restricted c...
ss2rabdv 3256 Deduction of restricted ab...
ss2rabi 3257 Inference of restricted ab...
rabss2 3258 Subclass law for restricte...
ssab2 3259 Subclass relation for the ...
ssrab2 3260 Subclass relation for a re...
rabssab 3261 A restricted class is a su...
uniiunlem 3262 A subset relationship usef...
dfpss2 3263 Alternate definition of pr...
dfpss3 3264 Alternate definition of pr...
psseq1 3265 Equality theorem for prope...
psseq2 3266 Equality theorem for prope...
psseq1i 3267 An equality inference for ...
psseq2i 3268 An equality inference for ...
psseq12i 3269 An equality inference for ...
psseq1d 3270 An equality deduction for ...
psseq2d 3271 An equality deduction for ...
psseq12d 3272 An equality deduction for ...
pssss 3273 A proper subclass is a sub...
pssne 3274 Two classes in a proper su...
pssssd 3275 Deduce subclass from prope...
pssned 3276 Proper subclasses are uneq...
sspss 3277 Subclass in terms of prope...
pssirr 3278 Proper subclass is irrefle...
pssn2lp 3279 Proper subclass has no 2-c...
sspsstri 3280 Two ways of stating tricho...
ssnpss 3281 Partial trichotomy law for...
psstr 3282 Transitive law for proper ...
sspsstr 3283 Transitive law for subclas...
psssstr 3284 Transitive law for subclas...
psstrd 3285 Proper subclass inclusion ...
sspsstrd 3286 Transitivity involving sub...
psssstrd 3287 Transitivity involving sub...
npss 3288 A class is not a proper su...
difeq1 3289 Equality theorem for class...
difeq2 3290 Equality theorem for class...
difeq12 3291 Equality theorem for class...
difeq1i 3292 Inference adding differenc...
difeq2i 3293 Inference adding differenc...
difeq12i 3294 Equality inference for cla...
difeq1d 3295 Deduction adding differenc...
difeq2d 3296 Deduction adding differenc...
difeq12d 3297 Equality deduction for cla...
difeqri 3298 Inference from membership ...
nfdif 3299 Bound-variable hypothesis ...
eldifi 3300 Implication of membership ...
eldifn 3301 Implication of membership ...
elndif 3302 A set does not belong to a...
neldif 3303 Implication of membership ...
difdif 3304 Double class difference. ...
difss 3305 Subclass relationship for ...
difssd 3306 A difference of two classe...
difss2 3307 If a class is contained in...
difss2d 3308 If a class is contained in...
ssdifss 3309 Preservation of a subclass...
ddif 3310 Double complement under un...
ssconb 3311 Contraposition law for sub...
sscon 3312 Contraposition law for sub...
ssdif 3313 Difference law for subsets...
ssdifd 3314 If ` A ` is contained in `...
sscond 3315 If ` A ` is contained in `...
ssdifssd 3316 If ` A ` is contained in `...
ssdif2d 3317 If ` A ` is contained in `...
elun 3318 Expansion of membership in...
uneqri 3319 Inference from membership ...
unidm 3320 Idempotent law for union o...
uncom 3321 Commutative law for union ...
equncom 3322 If a class equals the unio...
equncomi 3323 Inference form of ~ equnco...
uneq1 3324 Equality theorem for union...
uneq2 3325 Equality theorem for the u...
uneq12 3326 Equality theorem for union...
uneq1i 3327 Inference adding union to ...
uneq2i 3328 Inference adding union to ...
uneq12i 3329 Equality inference for uni...
uneq1d 3330 Deduction adding union to ...
uneq2d 3331 Deduction adding union to ...
uneq12d 3332 Equality deduction for uni...
nfun 3333 Bound-variable hypothesis ...
unass 3334 Associative law for union ...
un12 3335 A rearrangement of union. ...
un23 3336 A rearrangement of union. ...
un4 3337 A rearrangement of the uni...
unundi 3338 Union distributes over its...
unundir 3339 Union distributes over its...
ssun1 3340 Subclass relationship for ...
ssun2 3341 Subclass relationship for ...
ssun3 3342 Subclass law for union of ...
ssun4 3343 Subclass law for union of ...
elun1 3344 Membership law for union o...
elun2 3345 Membership law for union o...
unss1 3346 Subclass law for union of ...
ssequn1 3347 A relationship between sub...
unss2 3348 Subclass law for union of ...
unss12 3349 Subclass law for union of ...
ssequn2 3350 A relationship between sub...
unss 3351 The union of two subclasse...
unssi 3352 An inference showing the u...
unssd 3353 A deduction showing the un...
unssad 3354 If ` ( A u. B ) ` is conta...
unssbd 3355 If ` ( A u. B ) ` is conta...
ssun 3356 A condition that implies i...
rexun 3357 Restricted existential qua...
ralunb 3358 Restricted quantification ...
ralun 3359 Restricted quantification ...
elin 3360 Expansion of membership in...
elin2 3361 Membership in a class defi...
elin3 3362 Membership in a class defi...
incom 3363 Commutative law for inters...
ineqri 3364 Inference from membership ...
ineq1 3365 Equality theorem for inter...
ineq2 3366 Equality theorem for inter...
ineq12 3367 Equality theorem for inter...
ineq1i 3368 Equality inference for int...
ineq2i 3369 Equality inference for int...
ineq12i 3370 Equality inference for int...
ineq1d 3371 Equality deduction for int...
ineq2d 3372 Equality deduction for int...
ineq12d 3373 Equality deduction for int...
ineqan12d 3374 Equality deduction for int...
dfss1 3375 A frequently-used variant ...
dfss5 3376 Another definition of subc...
nfin 3377 Bound-variable hypothesis ...
csbing 3378 Distribute proper substitu...
rabbi2dva 3379 Deduction from a wff to a ...
inidm 3380 Idempotent law for interse...
inass 3381 Associative law for inters...
in12 3382 A rearrangement of interse...
in32 3383 A rearrangement of interse...
in13 3384 A rearrangement of interse...
in31 3385 A rearrangement of interse...
inrot 3386 Rotate the intersection of...
in4 3387 Rearrangement of intersect...
inindi 3388 Intersection distributes o...
inindir 3389 Intersection distributes o...
sseqin2 3390 A relationship between sub...
inss1 3391 The intersection of two cl...
inss2 3392 The intersection of two cl...
ssin 3393 Subclass of intersection. ...
ssini 3394 An inference showing that ...
ssind 3395 A deduction showing that a...
ssrin 3396 Add right intersection to ...
sslin 3397 Add left intersection to s...
ss2in 3398 Intersection of subclasses...
ssinss1 3399 Intersection preserves sub...
inss 3400 Inclusion of an intersecti...
unabs 3401 Absorption law for union. ...
inabs 3402 Absorption law for interse...
nssinpss 3403 Negation of subclass expre...
nsspssun 3404 Negation of subclass expre...
dfss4 3405 Subclass defined in terms ...
dfun2 3406 An alternate definition of...
dfin2 3407 An alternate definition of...
difin 3408 Difference with intersecti...
dfun3 3409 Union defined in terms of ...
dfin3 3410 Intersection defined in te...
dfin4 3411 Alternate definition of th...
invdif 3412 Intersection with universa...
indif 3413 Intersection with class di...
indif2 3414 Bring an intersection in a...
indif1 3415 Bring an intersection in a...
indifcom 3416 Commutation law for inters...
indi 3417 Distributive law for inter...
undi 3418 Distributive law for union...
indir 3419 Distributive law for inter...
undir 3420 Distributive law for union...
unineq 3421 Infer equality from equali...
uneqin 3422 Equality of union and inte...
difundi 3423 Distributive law for class...
difundir 3424 Distributive law for class...
difindi 3425 Distributive law for class...
difindir 3426 Distributive law for class...
indifdir 3427 Distribute intersection ov...
undm 3428 De Morgan's law for union....
indm 3429 De Morgan's law for inters...
difun1 3430 A relationship involving d...
undif3 3431 An equality involving clas...
difin2 3432 Represent a set difference...
dif32 3433 Swap second and third argu...
difabs 3434 Absorption-like law for cl...
symdif1 3435 Two ways to express symmet...
symdif2 3436 Two ways to express symmet...
unab 3437 Union of two class abstrac...
inab 3438 Intersection of two class ...
difab 3439 Difference of two class ab...
notab 3440 A class builder defined by...
unrab 3441 Union of two restricted cl...
inrab 3442 Intersection of two restri...
inrab2 3443 Intersection with a restri...
difrab 3444 Difference of two restrict...
dfrab2 3445 Alternate definition of re...
dfrab3 3446 Alternate definition of re...
notrab 3447 Complementation of restric...
dfrab3ss 3448 Restricted class abstracti...
rabun2 3449 Abstraction restricted to ...
reuss2 3450 Transfer uniqueness to a s...
reuss 3451 Transfer uniqueness to a s...
reuun1 3452 Transfer uniqueness to a s...
reuun2 3453 Transfer uniqueness to a s...
reupick 3454 Restricted uniqueness "pic...
reupick3 3455 Restricted uniqueness "pic...
reupick2 3456 Restricted uniqueness "pic...
dfnul2 3459 Alternate definition of th...
dfnul3 3460 Alternate definition of th...
noel 3461 The empty set has no eleme...
n0i 3462 If a set has elements, it ...
ne0i 3463 If a set has elements, it ...
vn0 3464 The universal class is not...
n0f 3465 A nonempty class has at le...
n0 3466 A nonempty class has at le...
neq0 3467 A nonempty class has at le...
reximdva0 3468 Restricted existence deduc...
n0moeu 3469 A case of equivalence of "...
rex0 3470 Vacuous existential quanti...
eq0 3471 The empty set has no eleme...
eqv 3472 The universe contains ever...
0el 3473 Membership of the empty se...
abvor0 3474 The class builder of a wff...
abn0 3475 Nonempty class abstraction...
rabn0 3476 Non-empty restricted class...
rab0 3477 Any restricted class abstr...
rabeq0 3478 Condition for a restricted...
rabxm 3479 Law of excluded middle, in...
rabnc 3480 Law of noncontradiction, i...
un0 3481 The union of a class with ...
in0 3482 The intersection of a clas...
inv1 3483 The intersection of a clas...
unv 3484 The union of a class with ...
0ss 3485 The null set is a subset o...
ss0b 3486 Any subset of the empty se...
ss0 3487 Any subset of the empty se...
sseq0 3488 A subclass of an empty cla...
ssn0 3489 A class with a nonempty su...
abf 3490 A class builder with a fal...
eq0rdv 3491 Deduction rule for equalit...
un00 3492 Two classes are empty iff ...
vss 3493 Only the universal class h...
0pss 3494 The null set is a proper s...
npss0 3495 No set is a proper subset ...
pssv 3496 Any non-universal class is...
disj 3497 Two ways of saying that tw...
disjr 3498 Two ways of saying that tw...
disj1 3499 Two ways of saying that tw...
reldisj 3500 Two ways of saying that tw...
disj3 3501 Two ways of saying that tw...
disjne 3502 Members of disjoint sets a...
disjel 3503 A set can't belong to both...
disj2 3504 Two ways of saying that tw...
disj4 3505 Two ways of saying that tw...
ssdisj 3506 Intersection with a subcla...
disjpss 3507 A class is a proper subset...
undisj1 3508 The union of disjoint clas...
undisj2 3509 The union of disjoint clas...
ssindif0 3510 Subclass expressed in term...
inelcm 3511 The intersection of classe...
minel 3512 A minimum element of a cla...
undif4 3513 Distribute union over diff...
disjssun 3514 Subset relation for disjoi...
ssdif0 3515 Subclass expressed in term...
vdif0 3516 Universal class equality i...
pssdifn0 3517 A proper subclass has a no...
pssdif 3518 A proper subclass has a no...
ssnelpss 3519 A subclass missing a membe...
ssnelpssd 3520 Subclass inclusion with on...
pssnel 3521 A proper subclass has a me...
difin0ss 3522 Difference, intersection, ...
inssdif0 3523 Intersection, subclass, an...
difid 3524 The difference between a c...
difidALT 3525 The difference between a c...
dif0 3526 The difference between a c...
0dif 3527 The difference between the...
disjdif 3528 A class and its relative c...
difin0 3529 The difference of a class ...
undifv 3530 The union of a class and i...
undif1 3531 Absorption of difference b...
undif2 3532 Absorption of difference b...
undifabs 3533 Absorption of difference b...
inundif 3534 The intersection and class...
difun2 3535 Absorption of union by dif...
undif 3536 Union of complementary par...
ssdifin0 3537 A subset of a difference d...
ssdifeq0 3538 A class is a subclass of i...
ssundif 3539 A condition equivalent to ...
difcom 3540 Swap the arguments of a cl...
pssdifcom1 3541 Two ways to express overla...
pssdifcom2 3542 Two ways to express non-co...
difdifdir 3543 Distributive law for class...
uneqdifeq 3544 Two ways to say that ` A `...
r19.2z 3545 Theorem 19.2 of [Margaris]...
r19.2zb 3546 A response to the notion t...
r19.3rz 3547 Restricted quantification ...
r19.28z 3548 Restricted quantifier vers...
r19.3rzv 3549 Restricted quantification ...
r19.9rzv 3550 Restricted quantification ...
r19.28zv 3551 Restricted quantifier vers...
r19.37zv 3552 Restricted quantifier vers...
r19.45zv 3553 Restricted version of Theo...
r19.27z 3554 Restricted quantifier vers...
r19.27zv 3555 Restricted quantifier vers...
r19.36zv 3556 Restricted quantifier vers...
rzal 3557 Vacuous quantification is ...
rexn0 3558 Restricted existential qua...
ralidm 3559 Idempotent law for restric...
ral0 3560 Vacuous universal quantifi...
rgenz 3561 Generalization rule that e...
ralf0 3562 The quantification of a fa...
raaan 3563 Rearrange restricted quant...
raaanv 3564 Rearrange restricted quant...
sbss 3565 Set substitution into the ...
sbcss 3566 Distribute proper substitu...
dfif2 3569 An alternate definition of...
dfif6 3570 An alternate definition of...
ifeq1 3571 Equality theorem for condi...
ifeq2 3572 Equality theorem for condi...
iftrue 3573 Value of the conditional o...
iffalse 3574 Value of the conditional o...
ifnefalse 3575 When values are unequal, b...
ifsb 3576 Distribute a function over...
dfif3 3577 Alternate definition of th...
dfif4 3578 Alternate definition of th...
dfif5 3579 Alternate definition of th...
ifeq12 3580 Equality theorem for condi...
ifeq1d 3581 Equality deduction for con...
ifeq2d 3582 Equality deduction for con...
ifeq12d 3583 Equality deduction for con...
ifbi 3584 Equivalence theorem for co...
ifbid 3585 Equivalence deduction for ...
ifbieq2i 3586 Equivalence/equality infer...
ifbieq2d 3587 Equivalence/equality deduc...
ifbieq12i 3588 Equivalence deduction for ...
ifbieq12d 3589 Equivalence deduction for ...
nfifd 3590 Deduction version of ~ nfi...
nfif 3591 Bound-variable hypothesis ...
ifeq1da 3592 Conditional equality. (Co...
ifeq2da 3593 Conditional equality. (Co...
ifclda 3594 Conditional closure. (Con...
csbifg 3595 Distribute proper substitu...
elimif 3596 Elimination of a condition...
ifbothda 3597 A wff ` th ` containing a ...
ifboth 3598 A wff ` th ` containing a ...
ifid 3599 Identical true and false a...
eqif 3600 Expansion of an equality w...
elif 3601 Membership in a conditiona...
ifel 3602 Membership of a conditiona...
ifcl 3603 Membership (closure) of a ...
ifeqor 3604 The possible values of a c...
ifnot 3605 Negating the first argumen...
ifan 3606 Rewrite a conjunction in a...
ifor 3607 Rewrite a disjunction in a...
dedth 3608 Weak deduction theorem tha...
dedth2h 3609 Weak deduction theorem eli...
dedth3h 3610 Weak deduction theorem eli...
dedth4h 3611 Weak deduction theorem eli...
dedth2v 3612 Weak deduction theorem for...
dedth3v 3613 Weak deduction theorem for...
dedth4v 3614 Weak deduction theorem for...
elimhyp 3615 Eliminate a hypothesis con...
elimhyp2v 3616 Eliminate a hypothesis con...
elimhyp3v 3617 Eliminate a hypothesis con...
elimhyp4v 3618 Eliminate a hypothesis con...
elimel 3619 Eliminate a membership hyp...
elimdhyp 3620 Version of ~ elimhyp where...
keephyp 3621 Transform a hypothesis ` p...
keephyp2v 3622 Keep a hypothesis containi...
keephyp3v 3623 Keep a hypothesis containi...
keepel 3624 Keep a membership hypothes...
ifex 3625 Conditional operator exist...
ifexg 3626 Conditional operator exist...
pwjust 3628 Soundness justification th...
pweq 3630 Equality theorem for power...
pweqi 3631 Equality inference for pow...
pweqd 3632 Equality deduction for pow...
elpw 3633 Membership in a power clas...
elpwg 3634 Membership in a power clas...
elpwi 3635 Subset relation implied by...
elpwid 3636 An element of a power clas...
elelpwi 3637 If ` A ` belongs to a part...
nfpw 3638 Bound-variable hypothesis ...
pwidg 3639 Membership of the original...
pwid 3640 A set is a member of its p...
pwss 3641 Subclass relationship for ...
snjust 3647 Soundness justification th...
sneq 3653 Equality theorem for singl...
sneqi 3654 Equality inference for sin...
sneqd 3655 Equality deduction for sin...
dfsn2 3656 Alternate definition of si...
elsn 3657 There is only one element ...
dfpr2 3658 Alternate definition of un...
elprg 3659 A member of an unordered p...
elpr 3660 A member of an unordered p...
elpr2 3661 A member of an unordered p...
elpri 3662 If a class is an element o...
nelpri 3663 If an element doesn't matc...
elsncg 3664 There is only one element ...
elsnc 3665 There is only one element ...
elsni 3666 There is only one element ...
snidg 3667 A set is a member of its s...
snidb 3668 A class is a set iff it is...
snid 3669 A set is a member of its s...
elsnc2g 3670 There is only one element ...
elsnc2 3671 There is only one element ...
ralsns 3672 Substitution expressed in ...
rexsns 3673 Restricted existential qua...
ralsng 3674 Substitution expressed in ...
rexsng 3675 Restricted existential qua...
ralsn 3676 Convert a quantification o...
rexsn 3677 Restricted existential qua...
eltpg 3678 Members of an unordered tr...
eltpi 3679 A member of an unordered t...
eltp 3680 A member of an unordered t...
dftp2 3681 Alternate definition of un...
nfpr 3682 Bound-variable hypothesis ...
ifpr 3683 Membership of a conditiona...
ralprg 3684 Convert a quantification o...
rexprg 3685 Convert a quantification o...
raltpg 3686 Convert a quantification o...
rextpg 3687 Convert a quantification o...
ralpr 3688 Convert a quantification o...
rexpr 3689 Convert an existential qua...
raltp 3690 Convert a quantification o...
rextp 3691 Convert a quantification o...
sbcsng 3692 Substitution expressed in ...
nfsn 3693 Bound-variable hypothesis ...
csbsng 3694 Distribute proper substitu...
disjsn 3695 Intersection with the sing...
disjsn2 3696 Intersection of distinct s...
snprc 3697 The singleton of a proper ...
r19.12sn 3698 Special case of ~ r19.12 w...
rabsn 3699 Condition where a restrict...
euabsn2 3700 Another way to express exi...
euabsn 3701 Another way to express exi...
reusn 3702 A way to express restricte...
absneu 3703 Restricted existential uni...
rabsneu 3704 Restricted existential uni...
eusn 3705 Two ways to express " ` A ...
rabsnt 3706 Truth implied by equality ...
prcom 3707 Commutative law for unorde...
preq1 3708 Equality theorem for unord...
preq2 3709 Equality theorem for unord...
preq12 3710 Equality theorem for unord...
preq1i 3711 Equality inference for uno...
preq2i 3712 Equality inference for uno...
preq12i 3713 Equality inference for uno...
preq1d 3714 Equality deduction for uno...
preq2d 3715 Equality deduction for uno...
preq12d 3716 Equality deduction for uno...
tpeq1 3717 Equality theorem for unord...
tpeq2 3718 Equality theorem for unord...
tpeq3 3719 Equality theorem for unord...
tpeq1d 3720 Equality theorem for unord...
tpeq2d 3721 Equality theorem for unord...
tpeq3d 3722 Equality theorem for unord...
tpeq123d 3723 Equality theorem for unord...
tprot 3724 Rotation of the elements o...
tpcoma 3725 Swap 1st and 2nd members o...
tpcomb 3726 Swap 2nd and 3rd members o...
tpass 3727 Split off the first elemen...
qdass 3728 Two ways to write an unord...
qdassr 3729 Two ways to write an unord...
tpidm12 3730 Unordered triple ` { A , A...
tpidm13 3731 Unordered triple ` { A , B...
tpidm23 3732 Unordered triple ` { A , B...
tpidm 3733 Unordered triple ` { A , A...
prid1g 3734 An unordered pair contains...
prid2g 3735 An unordered pair contains...
prid1 3736 An unordered pair contains...
prid2 3737 An unordered pair contains...
prprc1 3738 A proper class vanishes in...
prprc2 3739 A proper class vanishes in...
prprc 3740 An unordered pair containi...
tpid1 3741 One of the three elements ...
tpid2 3742 One of the three elements ...
tpid3g 3743 Closed theorem form of ~ t...
tpid3 3744 One of the three elements ...
snnzg 3745 The singleton of a set is ...
snnz 3746 The singleton of a set is ...
prnz 3747 A pair containing a set is...
prnzg 3748 A pair containing a set is...
tpnz 3749 A triplet containing a set...
snss 3750 The singleton of an elemen...
eldifsn 3751 Membership in a set with a...
eldifsni 3752 Membership in a set with a...
neldifsn 3753 ` A ` is not in ` ( B \ { ...
neldifsnd 3754 ` A ` is not in ` ( B \ { ...
rexdifsn 3755 Restricted existential qua...
snssg 3756 The singleton of an elemen...
difsn 3757 An element not in a set ca...
difprsn 3758 Removal of a singleton fro...
difsneq 3759 ` ( B \ { A } ) ` equals `...
difsnpss 3760 ` ( B \ { A } ) ` is a pro...
snssi 3761 The singleton of an elemen...
snssd 3762 The singleton of an elemen...
difsnid 3763 If we remove a single elem...
pw0 3764 Compute the power set of t...
pwpw0 3765 Compute the power set of t...
snsspr1 3766 A singleton is a subset of...
snsspr2 3767 A singleton is a subset of...
snsstp1 3768 A singleton is a subset of...
snsstp2 3769 A singleton is a subset of...
snsstp3 3770 A singleton is a subset of...
prss 3771 A pair of elements of a cl...
prssg 3772 A pair of elements of a cl...
prssi 3773 A pair of elements of a cl...
sssn 3774 The subsets of a singleton...
ssunsn2 3775 The property of being sand...
ssunsn 3776 Possible values for a set ...
eqsn 3777 Two ways to express that a...
ssunpr 3778 Possible values for a set ...
sspr 3779 The subsets of a pair. (C...
sstp 3780 The subsets of a triple. ...
tpss 3781 A triplet of elements of a...
sneqr 3782 If the singletons of two s...
snsssn 3783 If a singleton is a subset...
sneqrg 3784 Closed form of ~ sneqr . ...
sneqbg 3785 Two singletons of sets are...
snsspw 3786 The singleton of a class i...
prsspw 3787 An unordered pair belongs ...
preqr1 3788 Reverse equality lemma for...
preqr2 3789 Reverse equality lemma for...
preq12b 3790 Equality relationship for ...
prel12 3791 Equality of two unordered ...
opthpr 3792 A way to represent ordered...
preq12bg 3793 Closed form of ~ preq12b ....
preqsn 3794 Equivalence for a pair equ...
dfopif 3795 Rewrite ~ df-op using ` if...
dfopg 3796 Value of the ordered pair ...
dfop 3797 Value of an ordered pair w...
opeq1 3798 Equality theorem for order...
opeq2 3799 Equality theorem for order...
opeq12 3800 Equality theorem for order...
opeq1i 3801 Equality inference for ord...
opeq2i 3802 Equality inference for ord...
opeq12i 3803 Equality inference for ord...
opeq1d 3804 Equality deduction for ord...
opeq2d 3805 Equality deduction for ord...
opeq12d 3806 Equality deduction for ord...
oteq1 3807 Equality theorem for order...
oteq2 3808 Equality theorem for order...
oteq3 3809 Equality theorem for order...
oteq1d 3810 Equality deduction for ord...
oteq2d 3811 Equality deduction for ord...
oteq3d 3812 Equality deduction for ord...
oteq123d 3813 Equality deduction for ord...
nfop 3814 Bound-variable hypothesis ...
nfopd 3815 Deduction version of bound...
opid 3816 The ordered pair ` <. A , ...
ralunsn 3817 Restricted quantification ...
2ralunsn 3818 Double restricted quantifi...
opprc 3819 Expansion of an ordered pa...
opprc1 3820 Expansion of an ordered pa...
opprc2 3821 Expansion of an ordered pa...
oprcl 3822 If an ordered pair has an ...
pwsn 3823 The power set of a singlet...
pwsnALT 3824 The power set of a singlet...
pwpr 3825 The power set of an unorde...
pwtp 3826 The power set of an unorde...
pwpwpw0 3827 Compute the power set of t...
pwv 3828 The power class of the uni...
dfuni2 3831 Alternate definition of cl...
eluni 3832 Membership in class union....
eluni2 3833 Membership in class union....
elunii 3834 Membership in class union....
nfuni 3835 Bound-variable hypothesis ...
nfunid 3836 Deduction version of ~ nfu...
csbunig 3837 Distribute proper substitu...
unieq 3838 Equality theorem for class...
unieqi 3839 Inference of equality of t...
unieqd 3840 Deduction of equality of t...
eluniab 3841 Membership in union of a c...
elunirab 3842 Membership in union of a c...
unipr 3843 The union of a pair is the...
uniprg 3844 The union of a pair is the...
unisn 3845 A set equals the union of ...
unisng 3846 A set equals the union of ...
dfnfc2 3847 An alternative statement o...
uniun 3848 The class union of the uni...
uniin 3849 The class union of the int...
uniss 3850 Subclass relationship for ...
ssuni 3851 Subclass relationship for ...
unissi 3852 Subclass relationship for ...
unissd 3853 Subclass relationship for ...
uni0b 3854 The union of a set is empt...
uni0c 3855 The union of a set is empt...
uni0 3856 The union of the empty set...
elssuni 3857 An element of a class is a...
unissel 3858 Condition turning a subcla...
unissb 3859 Relationship involving mem...
uniss2 3860 A subclass condition on th...
unidif 3861 If the difference ` A \ B ...
ssunieq 3862 Relationship implying unio...
unimax 3863 Any member of a class is t...
dfint2 3866 Alternate definition of cl...
inteq 3867 Equality law for intersect...
inteqi 3868 Equality inference for cla...
inteqd 3869 Equality deduction for cla...
elint 3870 Membership in class inters...
elint2 3871 Membership in class inters...
elintg 3872 Membership in class inters...
elinti 3873 Membership in class inters...
nfint 3874 Bound-variable hypothesis ...
elintab 3875 Membership in the intersec...
elintrab 3876 Membership in the intersec...
elintrabg 3877 Membership in the intersec...
int0 3878 The intersection of the em...
intss1 3879 An element of a class incl...
ssint 3880 Subclass of a class inters...
ssintab 3881 Subclass of the intersecti...
ssintub 3882 Subclass of the least uppe...
ssmin 3883 Subclass of the minimum va...
intmin 3884 Any member of a class is t...
intss 3885 Intersection of subclasses...
intssuni 3886 The intersection of a none...
ssintrab 3887 Subclass of the intersecti...
unissint 3888 If the union of a class is...
intssuni2 3889 Subclass relationship for ...
intminss 3890 Under subset ordering, the...
intmin2 3891 Any set is the smallest of...
intmin3 3892 Under subset ordering, the...
intmin4 3893 Elimination of a conjunct ...
intab 3894 The intersection of a spec...
int0el 3895 The intersection of a clas...
intun 3896 The class intersection of ...
intpr 3897 The intersection of a pair...
intprg 3898 The intersection of a pair...
intsng 3899 Intersection of a singleto...
intsn 3900 The intersection of a sing...
uniintsn 3901 Two ways to express " ` A ...
uniintab 3902 The union and the intersec...
intunsn 3903 Theorem joining a singleto...
rint0 3904 Relative intersection of a...
elrint 3905 Membership in a restricted...
elrint2 3906 Membership in a restricted...
eliun 3911 Membership in indexed unio...
eliin 3912 Membership in indexed inte...
iuncom 3913 Commutation of indexed uni...
iuncom4 3914 Commutation of union with ...
iunconst 3915 Indexed union of a constan...
iinconst 3916 Indexed intersection of a ...
iuniin 3917 Law combining indexed unio...
iunss1 3918 Subclass theorem for index...
iinss1 3919 Subclass theorem for index...
iuneq1 3920 Equality theorem for index...
iineq1 3921 Equality theorem for restr...
ss2iun 3922 Subclass theorem for index...
iuneq2 3923 Equality theorem for index...
iineq2 3924 Equality theorem for index...
iuneq2i 3925 Equality inference for ind...
iineq2i 3926 Equality inference for ind...
iineq2d 3927 Equality deduction for ind...
iuneq2dv 3928 Equality deduction for ind...
iineq2dv 3929 Equality deduction for ind...
iuneq1d 3930 Equality theorem for index...
iuneq12d 3931 Equality deduction for ind...
iuneq2d 3932 Equality deduction for ind...
nfiun 3933 Bound-variable hypothesis ...
nfiin 3934 Bound-variable hypothesis ...
nfiu1 3935 Bound-variable hypothesis ...
nfii1 3936 Bound-variable hypothesis ...
dfiun2g 3937 Alternate definition of in...
dfiin2g 3938 Alternate definition of in...
dfiun2 3939 Alternate definition of in...
dfiin2 3940 Alternate definition of in...
cbviun 3941 Rule used to change the bo...
cbviin 3942 Change bound variables in ...
cbviunv 3943 Rule used to change the bo...
cbviinv 3944 Change bound variables in ...
iunss 3945 Subset theorem for an inde...
ssiun 3946 Subset implication for an ...
ssiun2 3947 Identity law for subset of...
ssiun2s 3948 Subset relationship for an...
iunss2 3949 A subclass condition on th...
iunab 3950 The indexed union of a cla...
iunrab 3951 The indexed union of a res...
iunxdif2 3952 Indexed union with a class...
ssiinf 3953 Subset theorem for an inde...
ssiin 3954 Subset theorem for an inde...
iinss 3955 Subset implication for an ...
iinss2 3956 An indexed intersection is...
uniiun 3957 Class union in terms of in...
intiin 3958 Class intersection in term...
iunid 3959 An indexed union of single...
iun0 3960 An indexed union of the em...
0iun 3961 An empty indexed union is ...
0iin 3962 An empty indexed intersect...
viin 3963 Indexed intersection with ...
iunn0 3964 There is a non-empty class...
iinab 3965 Indexed intersection of a ...
iinrab 3966 Indexed intersection of a ...
iinrab2 3967 Indexed intersection of a ...
iunin2 3968 Indexed union of intersect...
iunin1 3969 Indexed union of intersect...
iinun2 3970 Indexed intersection of un...
iundif2 3971 Indexed union of class dif...
2iunin 3972 Rearrange indexed unions o...
iindif2 3973 Indexed intersection of cl...
iinin2 3974 Indexed intersection of in...
iinin1 3975 Indexed intersection of in...
elriin 3976 Elementhood in a relative ...
riin0 3977 Relative intersection of a...
riinn0 3978 Relative intersection of a...
riinrab 3979 Relative intersection of a...
iinxsng 3980 A singleton index picks ou...
iinxprg 3981 Indexed intersection with ...
iunxsng 3982 A singleton index picks ou...
iunxsn 3983 A singleton index picks ou...
iunun 3984 Separate a union in an ind...
iunxun 3985 Separate a union in the in...
iunxiun 3986 Separate an indexed union ...
iinuni 3987 A relationship involving u...
iununi 3988 A relationship involving u...
sspwuni 3989 Subclass relationship for ...
pwssb 3990 Two ways to express a coll...
elpwuni 3991 Relationship for power cla...
iinpw 3992 The power class of an inte...
iunpwss 3993 Inclusion of an indexed un...
rintn0 3994 Relative intersection of a...
dfdisj2 3997 Alternate definition for d...
disjss2 3998 If each element of a colle...
disjeq2 3999 Equality theorem for disjo...
disjeq2dv 4000 Equality deduction for dis...
disjss1 4001 A subset of a disjoint col...
disjeq1 4002 Equality theorem for disjo...
disjeq1d 4003 Equality theorem for disjo...
disjeq12d 4004 Equality theorem for disjo...
cbvdisj 4005 Change bound variables in ...
cbvdisjv 4006 Change bound variables in ...
nfdisj 4007 Bound-variable hypothesis ...
nfdisj1 4008 Bound-variable hypothesis ...
disjor 4009 Two ways to say that a col...
disjmoOLD 4010 Two ways to say that a col...
disjors 4011 Two ways to say that a col...
disji2 4012 Property of a disjoint col...
disji 4013 Property of a disjoint col...
invdisj 4014 If there is a function ` C...
disjiun 4015 A disjoint collection yiel...
disjiunOLD 4016 A disjoint collection yiel...
sndisj 4017 Any collection of singleto...
0disj 4018 Any collection of empty se...
disjxsn 4019 A singleton collection is ...
disjx0 4020 An empty collection is dis...
disjprg 4021 A pair collection is disjo...
disjxiun 4022 An indexed union of a disj...
disjxun 4023 The union of two disjoint ...
disjss3 4024 Expand a disjoint collecti...
breq 4027 Equality theorem for binar...
breq1 4028 Equality theorem for a bin...
breq2 4029 Equality theorem for a bin...
breq12 4030 Equality theorem for a bin...
breqi 4031 Equality inference for bin...
breq1i 4032 Equality inference for a b...
breq2i 4033 Equality inference for a b...
breq12i 4034 Equality inference for a b...
breq1d 4035 Equality deduction for a b...
breqd 4036 Equality deduction for a b...
breq2d 4037 Equality deduction for a b...
breq12d 4038 Equality deduction for a b...
breq123d 4039 Equality deduction for a b...
breqan12d 4040 Equality deduction for a b...
breqan12rd 4041 Equality deduction for a b...
nbrne1 4042 Two classes are different ...
nbrne2 4043 Two classes are different ...
eqbrtri 4044 Substitution of equal clas...
eqbrtrd 4045 Substitution of equal clas...
eqbrtrri 4046 Substitution of equal clas...
eqbrtrrd 4047 Substitution of equal clas...
breqtri 4048 Substitution of equal clas...
breqtrd 4049 Substitution of equal clas...
breqtrri 4050 Substitution of equal clas...
breqtrrd 4051 Substitution of equal clas...
3brtr3i 4052 Substitution of equality i...
3brtr4i 4053 Substitution of equality i...
3brtr3d 4054 Substitution of equality i...
3brtr4d 4055 Substitution of equality i...
3brtr3g 4056 Substitution of equality i...
3brtr4g 4057 Substitution of equality i...
syl5eqbr 4058 B chained equality inferen...
syl5eqbrr 4059 B chained equality inferen...
syl5breq 4060 B chained equality inferen...
syl5breqr 4061 B chained equality inferen...
syl6eqbr 4062 A chained equality inferen...
syl6eqbrr 4063 A chained equality inferen...
syl6breq 4064 A chained equality inferen...
syl6breqr 4065 A chained equality inferen...
ssbrd 4066 Deduction from a subclass ...
ssbri 4067 Inference from a subclass ...
nfbrd 4068 Deduction version of bound...
nfbr 4069 Bound-variable hypothesis ...
brab1 4070 Relationship between a bin...
brun 4071 The union of two binary re...
brin 4072 The intersection of two re...
brdif 4073 The difference of two bina...
sbcbrg 4074 Move substitution in and o...
sbcbr12g 4075 Move substitution in and o...
sbcbr1g 4076 Move substitution in and o...
sbcbr2g 4077 Move substitution in and o...
opabss 4082 The collection of ordered ...
opabbid 4083 Equivalent wff's yield equ...
opabbidv 4084 Equivalent wff's yield equ...
opabbii 4085 Equivalent wff's yield equ...
nfopab 4086 Bound-variable hypothesis ...
nfopab1 4087 The first abstraction vari...
nfopab2 4088 The second abstraction var...
cbvopab 4089 Rule used to change bound ...
cbvopabv 4090 Rule used to change bound ...
cbvopab1 4091 Change first bound variabl...
cbvopab2 4092 Change second bound variab...
cbvopab1s 4093 Change first bound variabl...
cbvopab1v 4094 Rule used to change the fi...
cbvopab2v 4095 Rule used to change the se...
csbopabg 4096 Move substitution into a c...
unopab 4097 Union of two ordered pair ...
mpteq12f 4098 An equality theorem for th...
mpteq12dva 4099 An equality inference for ...
mpteq12dv 4100 An equality inference for ...
mpteq12 4101 An equality theorem for th...
mpteq1 4102 An equality theorem for th...
mpteq1d 4103 An equality theorem for th...
mpteq2ia 4104 An equality inference for ...
mpteq2i 4105 An equality inference for ...
mpteq12i 4106 An equality inference for ...
mpteq2da 4107 Slightly more general equa...
mpteq2dva 4108 Slightly more general equa...
mpteq2dv 4109 An equality inference for ...
nfmpt 4110 Bound-variable hypothesis ...
nfmpt1 4111 Bound-variable hypothesis ...
cbvmpt 4112 Rule to change the bound v...
cbvmptv 4113 Rule to change the bound v...
mptv 4114 Function with universal do...
dftr2 4117 An alternate way of defini...
dftr5 4118 An alternate way of defini...
dftr3 4119 An alternate way of defini...
dftr4 4120 An alternate way of defini...
treq 4121 Equality theorem for the t...
trel 4122 In a transitive class, the...
trel3 4123 In a transitive class, the...
trss 4124 An element of a transitive...
trin 4125 The intersection of transi...
tr0 4126 The empty set is transitiv...
trv 4127 The universe is transitive...
triun 4128 The indexed union of a cla...
truni 4129 The union of a class of tr...
trint 4130 The intersection of a clas...
trintss 4131 If ` A ` is transitive and...
trint0 4132 Any non-empty transitive c...
axrep1 4134 The version of the Axiom o...
axrep2 4135 Axiom of Replacement expre...
axrep3 4136 Axiom of Replacement sligh...
axrep4 4137 A more traditional version...
axrep5 4138 Axiom of Replacement (simi...
zfrepclf 4139 An inference rule based on...
zfrep3cl 4140 An inference rule based on...
zfrep4 4141 A version of Replacement u...
axsep 4142 Separation Scheme, which i...
axsep2 4144 A less restrictive version...
zfauscl 4145 Separation Scheme (Aussond...
bm1.3ii 4146 Convert implication to equ...
ax9vsep 4147 Derive a weakened version ...
zfnuleu 4148 Show the uniqueness of the...
axnulALT 4149 Prove ~ axnul directly fro...
axnul 4150 The Null Set Axiom of ZF s...
0ex 4152 The Null Set Axiom of ZF s...
nalset 4153 No set contains all sets. ...
vprc 4154 The universal class is not...
nvel 4155 The universal class doesn'...
vnex 4156 The universal class does n...
inex1 4157 Separation Scheme (Aussond...
inex2 4158 Separation Scheme (Aussond...
inex1g 4159 Closed-form, generalized S...
ssex 4160 The subset of a set is als...
ssexi 4161 The subset of a set is als...
ssexg 4162 The subset of a set is als...
ssexd 4163 A subclass of a set is a s...
difexg 4164 Existence of a difference....
zfausab 4165 Separation Scheme (Aussond...
rabexg 4166 Separation Scheme in terms...
rabex 4167 Separation Scheme in terms...
elssabg 4168 Membership in a class abst...
intex 4169 The intersection of a non-...
intnex 4170 If a class intersection is...
intexab 4171 The intersection of a non-...
intexrab 4172 The intersection of a non-...
iinexg 4173 The existence of an indexe...
intabs 4174 Absorption of a redundant ...
inuni 4175 The intersection of a unio...
elpw2g 4176 Membership in a power clas...
elpw2 4177 Membership in a power clas...
pwnss 4178 The power set of a set is ...
pwne 4179 No set equals its power se...
class2set 4180 Construct, from any class ...
class2seteq 4181 Equality theorem based on ...
0elpw 4182 Every power class contains...
0nep0 4183 The empty set and its powe...
0inp0 4184 Something cannot be equal ...
unidif0 4185 The removal of the empty s...
iin0 4186 An indexed intersection of...
notzfaus 4187 In the Separation Scheme ~...
intv 4188 The intersection of the un...
axpweq 4189 Two equivalent ways to exp...
zfpow 4191 Axiom of Power Sets expres...
axpow2 4192 A variant of the Axiom of ...
axpow3 4193 A variant of the Axiom of ...
el 4194 Every set is an element of...
pwex 4195 Power set axiom expressed ...
pwexg 4196 Power set axiom expressed ...
abssexg 4197 Existence of a class of su...
snexALT 4198 A singleton is a set. The...
p0ex 4199 The power set of the empty...
p0exALT 4200 The power set of the empty...
pp0ex 4201 The power set of the power...
ord3ex 4202 The ordinal number 3 is a ...
dtru 4203 At least two sets exist (o...
ax16b 4204 This theorem shows that ax...
eunex 4205 Existential uniqueness imp...
nfnid 4206 A set variable is not free...
nfcvb 4207 The "distinctor" expressio...
pwuni 4208 A class is a subclass of t...
dtruALT 4209 A version of ~ dtru ("two ...
dtrucor 4210 Corollary of ~ dtru . Thi...
dtrucor2 4211 The theorem form of the de...
dvdemo1 4212 Demonstration of a theorem...
dvdemo2 4213 Demonstration of a theorem...
zfpair 4214 The Axiom of Pairing of Ze...
axpr 4215 Unabbreviated version of t...
zfpair2 4217 Derive the abbreviated ver...
snex 4218 A singleton is a set. The...
prex 4219 The Axiom of Pairing using...
elALT 4220 Every set is an element of...
dtruALT2 4221 An alternative proof of ~ ...
snelpwi 4222 A singleton of a set belon...
snelpw 4223 A singleton of a set belon...
rext 4224 A theorem similar to exten...
sspwb 4225 Classes are subclasses if ...
unipw 4226 A class equals the union o...
pwel 4227 Membership of a power clas...
pwtr 4228 A class is transitive iff ...
ssextss 4229 An extensionality-like pri...
ssext 4230 An extensionality-like pri...
nssss 4231 Negation of subclass relat...
pweqb 4232 Classes are equal if and o...
intid 4233 The intersection of all se...
moabex 4234 "At most one" existence im...
rmorabex 4235 Restricted "at most one" e...
euabex 4236 The abstraction of a wff w...
nnullss 4237 A non-empty class (even if...
exss 4238 Restricted existence in a ...
opex 4239 An ordered pair of classes...
otex 4240 An ordered triple of class...
elop 4241 An ordered pair has two el...
opi1 4242 One of the two elements in...
opi2 4243 One of the two elements of...
opnz 4244 An ordered pair is nonempt...
opnzi 4245 An ordered pair is nonempt...
opth1 4246 Equality of the first memb...
opth 4247 The ordered pair theorem. ...
opthg 4248 Ordered pair theorem. ` C ...
opthg2 4249 Ordered pair theorem. (Co...
opth2 4250 Ordered pair theorem. (Co...
otth2 4251 Ordered triple theorem, wi...
otth 4252 Ordered triple theorem. (...
eqvinop 4253 A variable introduction la...
copsexg 4254 Substitution of class ` A ...
copsex2t 4255 Closed theorem form of ~ c...
copsex2g 4256 Implicit substitution infe...
copsex4g 4257 An implicit substitution i...
0nelop 4258 A property of ordered pair...
opeqex 4259 Equivalence of existence i...
oteqex2 4260 Equivalence of existence i...
oteqex 4261 Equivalence of existence i...
opcom 4262 An ordered pair commutes i...
moop2 4263 "At most one" property of ...
opeqsn 4264 Equivalence for an ordered...
opeqpr 4265 Equivalence for an ordered...
mosubopt 4266 "At most one" remains true...
mosubop 4267 "At most one" remains true...
euop2 4268 Transfer existential uniqu...
euotd 4269 Prove existential uniquene...
opthwiener 4270 Justification theorem for ...
uniop 4271 The union of an ordered pa...
uniopel 4272 Ordered pair membership is...
opabid 4273 The law of concretion. Sp...
elopab 4274 Membership in a class abst...
opelopabsbOLD 4275 The law of concretion in t...
brabsbOLD 4276 The law of concretion in t...
opelopabsb 4277 The law of concretion in t...
brabsb 4278 The law of concretion in t...
opelopabt 4279 Closed theorem form of ~ o...
opelopabga 4280 The law of concretion. Th...
brabga 4281 The law of concretion for ...
opelopab2a 4282 Ordered pair membership in...
opelopaba 4283 The law of concretion. Th...
braba 4284 The law of concretion for ...
opelopabg 4285 The law of concretion. Th...
brabg 4286 The law of concretion for ...
opelopab2 4287 Ordered pair membership in...
opelopab 4288 The law of concretion. Th...
brab 4289 The law of concretion for ...
opelopabaf 4290 The law of concretion. Th...
opelopabf 4291 The law of concretion. Th...
ssopab2 4292 Equivalence of ordered pai...
ssopab2b 4293 Equivalence of ordered pai...
ssopab2i 4294 Inference of ordered pair ...
ssopab2dv 4295 Inference of ordered pair ...
eqopab2b 4296 Equivalence of ordered pai...
opabn0 4297 Non-empty ordered pair cla...
iunopab 4298 Move indexed union inside ...
pwin 4299 The power class of the int...
pwunss 4300 The power class of the uni...
pwssun 4301 The power class of the uni...
pwundif 4302 Break up the power class o...
pwundifOLD 4303 Break up the power class o...
pwun 4304 The power class of the uni...
epelg 4308 The epsilon relation and m...
epelc 4309 The epsilon relationship a...
epel 4310 The epsilon relation and t...
dfid3 4312 A stronger version of ~ df...
dfid2 4313 Alternate definition of th...
poss 4318 Subset theorem for the par...
poeq1 4319 Equality theorem for parti...
poeq2 4320 Equality theorem for parti...
nfpo 4321 Bound-variable hypothesis ...
nfso 4322 Bound-variable hypothesis ...
pocl 4323 Properties of partial orde...
ispod 4324 Sufficient conditions for ...
swopolem 4325 Perform the substitutions ...
swopo 4326 A strict weak order is a p...
poirr 4327 A partial order relation i...
potr 4328 A partial order relation i...
po2nr 4329 A partial order relation h...
po3nr 4330 A partial order relation h...
po0 4331 Any relation is a partial ...
pofun 4332 A function preserves a par...
sopo 4333 A strict linear order is a...
soss 4334 Subset theorem for the str...
soeq1 4335 Equality theorem for the s...
soeq2 4336 Equality theorem for the s...
sonr 4337 A strict order relation is...
sotr 4338 A strict order relation is...
solin 4339 A strict order relation is...
so2nr 4340 A strict order relation ha...
so3nr 4341 A strict order relation ha...
sotric 4342 A strict order relation sa...
sotrieq 4343 Trichotomy law for strict ...
sotrieq2 4344 Trichotomy law for strict ...
sotr2 4345 A transitivity relation. ...
issod 4346 An irreflexive, transitive...
issoi 4347 An irreflexive, transitive...
isso2i 4348 Deduce strict ordering fro...
so0 4349 Any relation is a strict o...
somo 4350 A totally ordered set has ...
fri 4357 Property of well-founded r...
seex 4358 The ` R ` -preimage of an ...
exse 4359 Any relation on a set is s...
dffr2 4360 Alternate definition of we...
frc 4361 Property of well-founded r...
frss 4362 Subset theorem for the wel...
sess1 4363 Subset theorem for the set...
sess2 4364 Subset theorem for the set...
freq1 4365 Equality theorem for the w...
freq2 4366 Equality theorem for the w...
seeq1 4367 Equality theorem for the s...
seeq2 4368 Equality theorem for the s...
nffr 4369 Bound-variable hypothesis ...
nfse 4370 Bound-variable hypothesis ...
nfwe 4371 Bound-variable hypothesis ...
frirr 4372 A well-founded relation is...
fr2nr 4373 A well-founded relation ha...
fr0 4374 Any relation is well-found...
frminex 4375 If an element of a well-fo...
efrirr 4376 Irreflexivity of the epsil...
efrn2lp 4377 A set founded by epsilon c...
epse 4378 The epsilon relation is se...
tz7.2 4379 Similar to Theorem 7.2 of ...
dfepfr 4380 An alternate way of saying...
epfrc 4381 A subset of an epsilon-fou...
wess 4382 Subset theorem for the wel...
weeq1 4383 Equality theorem for the w...
weeq2 4384 Equality theorem for the w...
wefr 4385 A well-ordering is well-fo...
weso 4386 A well-ordering is a stric...
wecmpep 4387 The elements of an epsilon...
wetrep 4388 An epsilon well-ordering i...
wefrc 4389 A non-empty (possibly prop...
we0 4390 Any relation is a well-ord...
wereu 4391 A subset of a well-ordered...
wereu2 4392 All nonempty (possibly pro...
ordeq 4401 Equality theorem for the o...
elong 4402 An ordinal number is an or...
elon 4403 An ordinal number is an or...
eloni 4404 An ordinal number has the ...
elon2 4405 An ordinal number is an or...
limeq 4406 Equality theorem for the l...
ordwe 4407 Epsilon well-orders every ...
ordtr 4408 An ordinal class is transi...
ordfr 4409 Epsilon is well-founded on...
ordelss 4410 An element of an ordinal c...
trssord 4411 A transitive subclass of a...
ordirr 4412 Epsilon irreflexivity of o...
nordeq 4413 A member of an ordinal cla...
ordn2lp 4414 An ordinal class cannot an...
tz7.5 4415 A subclass (possibly prope...
ordelord 4416 An element of an ordinal c...
tron 4417 The class of all ordinal n...
ordelon 4418 An element of an ordinal c...
onelon 4419 An element of an ordinal n...
tz7.7 4420 Proposition 7.7 of [Takeut...
ordelssne 4421 Corollary 7.8 of [TakeutiZ...
ordelpss 4422 Corollary 7.8 of [TakeutiZ...
ordsseleq 4423 For ordinal classes, subcl...
ordin 4424 The intersection of two or...
onin 4425 The intersection of two or...
ordtri3or 4426 A trichotomy law for ordin...
ordtri1 4427 A trichotomy law for ordin...
ontri1 4428 A trichotomy law for ordin...
ordtri2 4429 A trichotomy law for ordin...
ordtri3 4430 A trichotomy law for ordin...
ordtri4 4431 A trichotomy law for ordin...
orddisj 4432 An ordinal class and its s...
onfr 4433 The ordinal class is well-...
onelpss 4434 Relationship between membe...
onsseleq 4435 Relationship between subse...
onelss 4436 An element of an ordinal n...
ordtr1 4437 Transitive law for ordinal...
ordtr2 4438 Transitive law for ordinal...
ordtr3 4439 Transitive law for ordinal...
ontr1 4440 Transitive law for ordinal...
ontr2 4441 Transitive law for ordinal...
ordunidif 4442 The union of an ordinal st...
ordintdif 4443 If ` B ` is smaller than `...
onintss 4444 If a property is true for ...
oneqmini 4445 A way to show that an ordi...
ord0 4446 The empty set is an ordina...
0elon 4447 The empty set is an ordina...
ord0eln0 4448 A non-empty ordinal contai...
on0eln0 4449 An ordinal number contains...
dflim2 4450 An alternate definition of...
inton 4451 The intersection of the cl...
nlim0 4452 The empty set is not a lim...
limord 4453 A limit ordinal is ordinal...
limuni 4454 A limit ordinal is its own...
limuni2 4455 The union of a limit ordin...
0ellim 4456 A limit ordinal contains t...
limelon 4457 A limit ordinal class that...
onn0 4458 The class of all ordinal n...
suceq 4459 Equality of successors. (...
elsuci 4460 Membership in a successor....
elsucg 4461 Membership in a successor....
elsuc2g 4462 Variant of membership in a...
elsuc 4463 Membership in a successor....
elsuc2 4464 Membership in a successor....
nfsuc 4465 Bound-variable hypothesis ...
elelsuc 4466 Membership in a successor....
sucel 4467 Membership of a successor ...
suc0 4468 The successor of the empty...
sucprc 4469 A proper class is its own ...
unisuc 4470 A transitive class is equa...
sssucid 4471 A class is included in its...
sucidg 4472 Part of Proposition 7.23 o...
sucid 4473 A set belongs to its succe...
nsuceq0 4474 No successor is empty. (C...
eqelsuc 4475 A set belongs to the succe...
iunsuc 4476 Inductive definition for t...
suctr 4477 The successor of a transti...
trsuc 4478 A set whose successor belo...
trsuc2OLD 4479 Obsolete proof of ~ suctr ...
trsucss 4480 A member of the successor ...
ordsssuc 4481 A subset of an ordinal bel...
onsssuc 4482 A subset of an ordinal num...
ordsssuc2 4483 An ordinal subset of an or...
onmindif 4484 When its successor is subt...
ordnbtwn 4485 There is no set between an...
onnbtwn 4486 There is no set between an...
sucssel 4487 A set whose successor is a...
orddif 4488 Ordinal derived from its s...
orduniss 4489 An ordinal class includes ...
ordtri2or 4490 A trichotomy law for ordin...
ordtri2or2 4491 A trichotomy law for ordin...
ordtri2or3 4492 A consequence of total ord...
ordelinel 4493 The intersection of two or...
ordssun 4494 Property of a subclass of ...
ordequn 4495 The maximum (i.e. union) o...
ordun 4496 The maximum (i.e. union) o...
ordunisssuc 4497 A subclass relationship fo...
suc11 4498 The successor operation be...
onordi 4499 An ordinal number is an or...
ontrci 4500 An ordinal number is a tra...
onirri 4501 An ordinal number is not a...
oneli 4502 A member of an ordinal num...
onelssi 4503 A member of an ordinal num...
onssneli 4504 An ordering law for ordina...
onssnel2i 4505 An ordering law for ordina...
onelini 4506 An element of an ordinal n...
oneluni 4507 An ordinal number equals i...
onunisuci 4508 An ordinal number is equal...
onsseli 4509 Subset is equivalent to me...
onun2i 4510 The union of two ordinal n...
unizlim 4511 An ordinal equal to its ow...
on0eqel 4512 An ordinal number either e...
snsn0non 4513 The singleton of the singl...
zfun 4515 Axiom of Union expressed w...
axun2 4516 A variant of the Axiom of ...
uniex2 4517 The Axiom of Union using t...
uniex 4518 The Axiom of Union in clas...
uniexg 4519 The ZF Axiom of Union in c...
unex 4520 The union of two sets is a...
tpex 4521 A triple of classes exists...
unexb 4522 Existence of union is equi...
unexg 4523 A union of two sets is a s...
unisn2 4524 A version of ~ unisn witho...
unisn3 4525 Union of a singleton in th...
snnex 4526 The class of all singleton...
difex2 4527 If the subtrahend of a cla...
opeluu 4528 Each member of an ordered ...
uniuni 4529 Expression for double unio...
eusv1 4530 Two ways to express single...
eusvnf 4531 Even if ` x ` is free in `...
eusvnfb 4532 Two ways to say that ` A (...
eusv2i 4533 Two ways to express single...
eusv2nf 4534 Two ways to express single...
eusv2 4535 Two ways to express single...
reusv1 4536 Two ways to express single...
reusv2lem1 4537 Lemma for ~ reusv2 . (Con...
reusv2lem2 4538 Lemma for ~ reusv2 . (Con...
reusv2lem3 4539 Lemma for ~ reusv2 . (Con...
reusv2lem4 4540 Lemma for ~ reusv2 . (Con...
reusv2lem5 4541 Lemma for ~ reusv2 . (Con...
reusv2 4542 Two ways to express single...
reusv3i 4543 Two ways of expressing exi...
reusv3 4544 Two ways to express single...
eusv4 4545 Two ways to express single...
reusv5OLD 4546 Two ways to express single...
reusv6OLD 4547 Two ways to express single...
reusv7OLD 4548 Two ways to express single...
alxfr 4549 Transfer universal quantif...
ralxfrd 4550 Transfer universal quantif...
rexxfrd 4551 Transfer universal quantif...
ralxfr2d 4552 Transfer universal quantif...
rexxfr2d 4553 Transfer universal quantif...
ralxfr 4554 Transfer universal quantif...
ralxfrALT 4555 Transfer universal quantif...
rexxfr 4556 Transfer existence from a ...
rabxfrd 4557 Class builder membership a...
rabxfr 4558 Class builder membership a...
reuxfr2d 4559 Transfer existential uniqu...
reuxfr2 4560 Transfer existential uniqu...
reuxfrd 4561 Transfer existential uniqu...
reuxfr 4562 Transfer existential uniqu...
reuhypd 4563 A theorem useful for elimi...
reuhyp 4564 A theorem useful for elimi...
uniexb 4565 The Axiom of Union and its...
pwexb 4566 The Axiom of Power Sets an...
univ 4567 The union of the universe ...
eldifpw 4568 Membership in a power clas...
elpwun 4569 Membership in the power cl...
elpwunsn 4570 Membership in an extension...
op1stb 4571 Extract the first member o...
iunpw 4572 An indexed union of a powe...
fr3nr 4573 A well-founded relation ha...
epne3 4574 A set well-founded by epsi...
dfwe2 4575 Alternate definition of we...
ordon 4576 The class of all ordinal n...
epweon 4577 The epsilon relation well-...
onprc 4578 No set contains all ordina...
ssorduni 4579 The union of a class of or...
ssonuni 4580 The union of a set of ordi...
ssonunii 4581 The union of a set of ordi...
ordeleqon 4582 A way to express the ordin...
ordsson 4583 Any ordinal class is a sub...
onss 4584 An ordinal number is a sub...
ssonprc 4585 Two ways of saying a class...
onuni 4586 The union of an ordinal nu...
orduni 4587 The union of an ordinal cl...
onint 4588 The intersection (infimum)...
onint0 4589 The intersection of a clas...
onssmin 4590 A non-empty class of ordin...
onminesb 4591 If a property is true for ...
onminsb 4592 If a property is true for ...
oninton 4593 The intersection of a non-...
onintrab 4594 The intersection of a clas...
onintrab2 4595 An existence condition equ...
onnmin 4596 No member of a set of ordi...
onnminsb 4597 An ordinal number smaller ...
oneqmin 4598 A way to show that an ordi...
bm2.5ii 4599 Problem 2.5(ii) of [BellMa...
onminex 4600 If a wff is true for an or...
sucon 4601 The class of all ordinal n...
sucexb 4602 A successor exists iff its...
sucexg 4603 The successor of a set is ...
sucex 4604 The successor of a set is ...
onmindif2 4605 The minimum of a class of ...
suceloni 4606 The successor of an ordina...
ordsuc 4607 The successor of an ordina...
ordpwsuc 4608 The collection of ordinals...
onpwsuc 4609 The collection of ordinal ...
sucelon 4610 The successor of an ordina...
ordsucss 4611 The successor of an elemen...
onpsssuc 4612 An ordinal number is a pro...
ordelsuc 4613 A set belongs to an ordina...
onsucmin 4614 The successor of an ordina...
ordsucelsuc 4615 Membership is inherited by...
ordsucsssuc 4616 The subclass relationship ...
ordsucuniel 4617 Given an element ` A ` of ...
ordsucun 4618 The successor of the maxim...
ordunpr 4619 The maximum of two ordinal...
ordunel 4620 The maximum of two ordinal...
onsucuni 4621 A class of ordinal numbers...
ordsucuni 4622 An ordinal class is a subc...
orduniorsuc 4623 An ordinal class is either...
unon 4624 The class of all ordinal n...
ordunisuc 4625 An ordinal class is equal ...
orduniss2 4626 The union of the ordinal s...
onsucuni2 4627 A successor ordinal is the...
0elsuc 4628 The successor of an ordina...
limon 4629 The class of ordinal numbe...
onssi 4630 An ordinal number is a sub...
onsuci 4631 The successor of an ordina...
onuniorsuci 4632 An ordinal number is eithe...
onuninsuci 4633 A limit ordinal is not a s...
onsucssi 4634 A set belongs to an ordina...
nlimsucg 4635 A successor is not a limit...
orduninsuc 4636 An ordinal equal to its un...
ordunisuc2 4637 An ordinal equal to its un...
ordzsl 4638 An ordinal is zero, a succ...
onzsl 4639 An ordinal number is zero,...
dflim3 4640 An alternate definition of...
dflim4 4641 An alternate definition of...
limsuc 4642 The successor of a member ...
limsssuc 4643 A class includes a limit o...
nlimon 4644 Two ways to express the cl...
limuni3 4645 The union of a nonempty cl...
tfi 4646 The Principle of Transfini...
tfis 4647 Transfinite Induction Sche...
tfis2f 4648 Transfinite Induction Sche...
tfis2 4649 Transfinite Induction Sche...
tfis3 4650 Transfinite Induction Sche...
tfisi 4651 A transfinite induction sc...
tfinds 4652 Principle of Transfinite I...
tfindsg 4653 Transfinite Induction (inf...
tfindsg2 4654 Transfinite Induction (inf...
tfindes 4655 Transfinite Induction with...
tfinds2 4656 Transfinite Induction (inf...
tfinds3 4657 Principle of Transfinite I...
dfom2 4660 An alternate definition of...
elom 4661 Membership in omega. The ...
omsson 4662 Omega is a subset of ` On ...
limomss 4663 The class of natural numbe...
nnon 4664 A natural number is an ord...
nnoni 4665 A natural number is an ord...
nnord 4666 A natural number is ordina...
ordom 4667 Omega is ordinal. Theorem...
elnn 4668 A member of a natural numb...
omon 4669 The class of natural numbe...
omelon2 4670 Omega is an ordinal number...
nnlim 4671 A natural number is not a ...
omssnlim 4672 The class of natural numbe...
limom 4673 Omega is a limit ordinal. ...
peano2b 4674 A class belongs to omega i...
nnsuc 4675 A nonzero natural number i...
ssnlim 4676 An ordinal subclass of non...
peano1 4677 Zero is a natural number. ...
peano2 4678 The successor of any natur...
peano3 4679 The successor of any natur...
peano4 4680 Two natural numbers are eq...
peano5 4681 The induction postulate: a...
nn0suc 4682 A natural number is either...
find 4683 The Principle of Finite In...
finds 4684 Principle of Finite Induct...
findsg 4685 Principle of Finite Induct...
finds2 4686 Principle of Finite Induct...
finds1 4687 Principle of Finite Induct...
findes 4688 Finite induction with expl...
xpeq1 4705 Equality theorem for cross...
xpeq2 4706 Equality theorem for cross...
elxpi 4707 Membership in a cross prod...
elxp 4708 Membership in a cross prod...
elxp2 4709 Membership in a cross prod...
xpeq12 4710 Equality theorem for cross...
xpeq1i 4711 Equality inference for cro...
xpeq2i 4712 Equality inference for cro...
xpeq12i 4713 Equality inference for cro...
xpeq1d 4714 Equality deduction for cro...
xpeq2d 4715 Equality deduction for cro...
xpeq12d 4716 Equality deduction for cro...
nfxp 4717 Bound-variable hypothesis ...
csbxpg 4718 Distribute proper substitu...
0nelxp 4719 The empty set is not a mem...
0nelelxp 4720 A member of a cross produc...
opelxp 4721 Ordered pair membership in...
brxp 4722 Binary relation on a cross...
opelxpi 4723 Ordered pair membership in...
opelxp1 4724 The first member of an ord...
opelxp2 4725 The second member of an or...
otelxp1 4726 The first member of an ord...
rabxp 4727 Membership in a class buil...
brrelex12 4728 A true binary relation on ...
brrelex 4729 A true binary relation on ...
brrelex2 4730 A true binary relation on ...
brrelexi 4731 The first argument of a bi...
brrelex2i 4732 The second argument of a b...
nprrel 4733 No proper class is related...
fconstmpt 4734 Representation of a consta...
vtoclr 4735 Variable to class conversi...
opelvvg 4736 Ordered pair membership in...
opelvv 4737 Ordered pair membership in...
opthprc 4738 Justification theorem for ...
brel 4739 Two things in a binary rel...
brab2a 4740 Ordered pair membership in...
elxp3 4741 Membership in a cross prod...
opeliunxp 4742 Membership in a union of c...
xpundi 4743 Distributive law for cross...
xpundir 4744 Distributive law for cross...
xpiundi 4745 Distributive law for cross...
xpiundir 4746 Distributive law for cross...
resiundiOLD 4747 Obsolete proof of ~ resiun...
iunxpconst 4748 Membership in a union of c...
xpun 4749 The cross product of two u...
elvv 4750 Membership in universal cl...
elvvv 4751 Membership in universal cl...
elvvuni 4752 An ordered pair contains i...
brinxp2 4753 Intersection of binary rel...
brinxp 4754 Intersection of binary rel...
poinxp 4755 Intersection of partial or...
soinxp 4756 Intersection of total orde...
frinxp 4757 Intersection of well-found...
seinxp 4758 Intersection of set-like r...
weinxp 4759 Intersection of well-order...
posn 4760 Partial ordering of a sing...
sosn 4761 Strict ordering on a singl...
frsn 4762 Founded relation on a sing...
wesn 4763 Well-ordering of a singlet...
opabssxp 4764 An abstraction relation is...
brab2ga 4765 The law of concretion for ...
optocl 4766 Implicit substitution of c...
2optocl 4767 Implicit substitution of c...
3optocl 4768 Implicit substitution of c...
opbrop 4769 Ordered pair membership in...
xp0r 4770 The cross product with the...
onxpdisj 4771 Ordinal numbers and ordere...
onnev 4772 The class of ordinal numbe...
releq 4773 Equality theorem for the r...
releqi 4774 Equality inference for the...
releqd 4775 Equality deduction for the...
nfrel 4776 Bound-variable hypothesis ...
relss 4777 Subclass theorem for relat...
ssrel 4778 A subclass relationship de...
eqrel 4779 Extensionality principle f...
relssi 4780 Inference from subclass pr...
relssdv 4781 Deduction from subclass pr...
eqrelriv 4782 Inference from extensional...
eqrelriiv 4783 Inference from extensional...
eqbrriv 4784 Inference from extensional...
eqrelrdv 4785 Deduce equality of relatio...
eqbrrdv 4786 Deduction from extensional...
eqbrrdiv 4787 Deduction from extensional...
eqrelrdv2 4788 A version of ~ eqrelrdv . ...
ssrelrel 4789 A subclass relationship de...
eqrelrel 4790 Extensionality principle f...
elrel 4791 A member of a relation is ...
relsn 4792 A singleton is a relation ...
relsnop 4793 A singleton of an ordered ...
xpss12 4794 Subset theorem for cross p...
xpss 4795 A cross product is include...
relxp 4796 A cross product is a relat...
xpss1 4797 Subset relation for cross ...
xpss2 4798 Subset relation for cross ...
xpsspw 4799 A cross product is include...
xpsspwOLD 4800 A cross product is include...
unixpss 4801 The double class union of ...
xpexg 4802 The cross product of two s...
xpex 4803 The cross product of two s...
relun 4804 The union of two relations...
relin1 4805 The intersection with a re...
relin2 4806 The intersection with a re...
reldif 4807 A difference cutting down ...
reliun 4808 An indexed union is a rela...
reliin 4809 An indexed intersection is...
reluni 4810 The union of a class is a ...
relint 4811 The intersection of a clas...
rel0 4812 The empty set is a relatio...
relopabi 4813 A class of ordered pairs i...
relopab 4814 A class of ordered pairs i...
reli 4815 The identity relation is a...
rele 4816 The membership relation is...
opabid2 4817 A relation expressed as an...
inopab 4818 Intersection of two ordere...
difopab 4819 The difference of two orde...
inxp 4820 The intersection of two cr...
xpindi 4821 Distributive law for cross...
xpindir 4822 Distributive law for cross...
xpiindi 4823 Distributive law for cross...
xpriindi 4824 Distributive law for cross...
eliunxp 4825 Membership in a union of c...
opeliunxp2 4826 Membership in a union of c...
raliunxp 4827 Write a double restricted ...
rexiunxp 4828 Write a double restricted ...
ralxp 4829 Universal quantification r...
rexxp 4830 Existential quantification...
djussxp 4831 Disjoint union is a subset...
ralxpf 4832 Version of ~ ralxp with bo...
rexxpf 4833 Version of ~ rexxp with bo...
iunxpf 4834 Indexed union on a cross p...
opabbi2dv 4835 Deduce equality of a relat...
relop 4836 A necessary and sufficient...
ideqg 4837 For sets, the identity rel...
ideq 4838 For sets, the identity rel...
ididg 4839 A set is identical to itse...
issetid 4840 Two ways of expressing set...
coss1 4841 Subclass theorem for compo...
coss2 4842 Subclass theorem for compo...
coeq1 4843 Equality theorem for compo...
coeq2 4844 Equality theorem for compo...
coeq1i 4845 Equality inference for com...
coeq2i 4846 Equality inference for com...
coeq1d 4847 Equality deduction for com...
coeq2d 4848 Equality deduction for com...
coeq12i 4849 Equality inference for com...
coeq12d 4850 Equality deduction for com...
nfco 4851 Bound-variable hypothesis ...
brcog 4852 Ordered pair membership in...
opelco2g 4853 Ordered pair membership in...
brco 4854 Binary relation on a compo...
opelco 4855 Ordered pair membership in...
cnvss 4856 Subset theorem for convers...
cnveq 4857 Equality theorem for conve...
cnveqi 4858 Equality inference for con...
cnveqd 4859 Equality deduction for con...
elcnv 4860 Membership in a converse. ...
elcnv2 4861 Membership in a converse. ...
nfcnv 4862 Bound-variable hypothesis ...
opelcnvg 4863 Ordered-pair membership in...
brcnvg 4864 The converse of a binary r...
opelcnv 4865 Ordered-pair membership in...
brcnv 4866 The converse of a binary r...
cnvco 4867 Distributive law of conver...
cnvuni 4868 The converse of a class un...
dfdm3 4869 Alternate definition of do...
dfrn2 4870 Alternate definition of ra...
dfrn3 4871 Alternate definition of ra...
elrn2g 4872 Membership in a range. (C...
elrng 4873 Membership in a range. (C...
dfdm4 4874 Alternate definition of do...
dfdmf 4875 Definition of domain, usin...
eldmg 4876 Domain membership. Theore...
eldm2g 4877 Domain membership. Theore...
eldm 4878 Membership in a domain. T...
eldm2 4879 Membership in a domain. T...
dmss 4880 Subset theorem for domain....
dmeq 4881 Equality theorem for domai...
dmeqi 4882 Equality inference for dom...
dmeqd 4883 Equality deduction for dom...
opeldm 4884 Membership of first of an ...
breldm 4885 Membership of first of a b...
breldmg 4886 Membership of first of a b...
dmun 4887 The domain of a union is t...
dmin 4888 The domain of an intersect...
dmiun 4889 The domain of an indexed u...
dmuni 4890 The domain of a union. Pa...
dmopab 4891 The domain of a class of o...
dmopabss 4892 Upper bound for the domain...
dmopab3 4893 The domain of a restricted...
dm0 4894 The domain of the empty se...
dmi 4895 The domain of the identity...
dmv 4896 The domain of the universe...
dm0rn0 4897 An empty domain implies an...
reldm0 4898 A relation is empty iff it...
dmxp 4899 The domain of a cross prod...
dmxpid 4900 The domain of a square cro...
dmxpin 4901 The domain of the intersec...
xpid11 4902 The cross product of a cla...
dmcnvcnv 4903 The domain of the double c...
rncnvcnv 4904 The range of the double co...
elreldm 4905 The first member of an ord...
rneq 4906 Equality theorem for range...
rneqi 4907 Equality inference for ran...
rneqd 4908 Equality deduction for ran...
rnss 4909 Subset theorem for range. ...
brelrng 4910 The second argument of a b...
brelrn 4911 The second argument of a b...
opelrn 4912 Membership of second membe...
releldm 4913 The first argument of a bi...
relelrn 4914 The second argument of a b...
releldmb 4915 Membership in a domain. (...
relelrnb 4916 Membership in a range. (C...
releldmi 4917 The first argument of a bi...
relelrni 4918 The second argument of a b...
dfrnf 4919 Definition of range, using...
elrn2 4920 Membership in a range. (C...
elrn 4921 Membership in a range. (C...
nfdm 4922 Bound-variable hypothesis ...
nfrn 4923 Bound-variable hypothesis ...
dmiin 4924 Domain of an intersection....
csbrng 4925 Distribute proper substitu...
rnopab 4926 The range of a class of or...
rnmpt 4927 The range of a function in...
elrnmpt 4928 The range of a function in...
elrnmpt1s 4929 Elementhood in an image se...
elrnmpt1 4930 Elementhood in an image se...
elrnmptg 4931 Membership in the range of...
elrnmpti 4932 Membership in the range of...
dfiun3g 4933 Alternate definition of in...
dfiin3g 4934 Alternate definition of in...
dfiun3 4935 Alternate definition of in...
dfiin3 4936 Alternate definition of in...
riinint 4937 Express a relative indexed...
rn0 4938 The range of the empty set...
relrn0 4939 A relation is empty iff it...
dmrnssfld 4940 The domain and range of a ...
dmexg 4941 The domain of a set is a s...
rnexg 4942 The range of a set is a se...
dmex 4943 The domain of a set is a s...
rnex 4944 The range of a set is a se...
iprc 4945 The identity function is a...
dmcoss 4946 Domain of a composition. ...
rncoss 4947 Range of a composition. (...
dmcosseq 4948 Domain of a composition. ...
dmcoeq 4949 Domain of a composition. ...
rncoeq 4950 Range of a composition. (...
reseq1 4951 Equality theorem for restr...
reseq2 4952 Equality theorem for restr...
reseq1i 4953 Equality inference for res...
reseq2i 4954 Equality inference for res...
reseq12i 4955 Equality inference for res...
reseq1d 4956 Equality deduction for res...
reseq2d 4957 Equality deduction for res...
reseq12d 4958 Equality deduction for res...
nfres 4959 Bound-variable hypothesis ...
csbresg 4960 Distribute proper substitu...
res0 4961 A restriction to the empty...
opelres 4962 Ordered pair membership in...
brres 4963 Binary relation on a restr...
opelresg 4964 Ordered pair membership in...
brresg 4965 Binary relation on a restr...
opres 4966 Ordered pair membership in...
resieq 4967 A restricted identity rela...
opelresiOLD 4968 ` <. A , A >. ` belongs to...
opelresi 4969 ` <. A , A >. ` belongs to...
resres 4970 The restriction of a restr...
resundi 4971 Distributive law for restr...
resundir 4972 Distributive law for restr...
resindi 4973 Class restriction distribu...
resindir 4974 Class restriction distribu...
inres 4975 Move intersection into cla...
resiun1 4976 Distribution of restrictio...
resiun2 4977 Distribution of restrictio...
dmres 4978 The domain of a restrictio...
ssdmres 4979 A domain restricted to a s...
dmresexg 4980 The domain of a restrictio...
resss 4981 A class includes its restr...
rescom 4982 Commutative law for restri...
ssres 4983 Subclass theorem for restr...
ssres2 4984 Subclass theorem for restr...
relres 4985 A restriction is a relatio...
resabs1 4986 Absorption law for restric...
resabs2 4987 Absorption law for restric...
residm 4988 Idempotent law for restric...
resima 4989 A restriction to an image....
resima2 4990 Image under a restricted c...
xpssres 4991 Restriction of a constant ...
elres 4992 Membership in a restrictio...
elsnres 4993 Memebership in restriction...
relssres 4994 Simplification law for res...
resdm 4995 A relation restricted to i...
resexg 4996 The restriction of a set i...
resex 4997 The restriction of a set i...
resopab 4998 Restriction of a class abs...
resiexg 4999 The existence of a restric...
iss 5000 A subclass of the identity...
resopab2 5001 Restriction of a class abs...
resmpt 5002 Restriction of the mapping...
resmpt3 5003 Unconditional restriction ...
dfres2 5004 Alternate definition of th...
opabresid 5005 The restricted identity ex...
mptresid 5006 The restricted identity ex...
dmresi 5007 The domain of a restricted...
resid 5008 Any relation restricted to...
imaeq1 5009 Equality theorem for image...
imaeq2 5010 Equality theorem for image...
imaeq1i 5011 Equality theorem for image...
imaeq2i 5012 Equality theorem for image...
imaeq1d 5013 Equality theorem for image...
imaeq2d 5014 Equality theorem for image...
imaeq12d 5015 Equality theorem for image...
dfima2 5016 Alternate definition of im...
dfima3 5017 Alternate definition of im...
elimag 5018 Membership in an image. T...
elima 5019 Membership in an image. T...
elima2 5020 Membership in an image. T...
elima3 5021 Membership in an image. T...
nfima 5022 Bound-variable hypothesis ...
nfimad 5023 Deduction version of bound...
csbima12g 5024 Move class substitution in...
csbima12gALT 5025 Move class substitution in...
imadmrn 5026 The image of the domain of...
imassrn 5027 The image of a class is a ...
imaexg 5028 The image of a set is a se...
imai 5029 Image under the identity r...
rnresi 5030 The range of the restricte...
resiima 5031 The image of a restriction...
ima0 5032 Image of the empty set. T...
0ima 5033 Image under the empty rela...
imadisj 5034 A class whose image under ...
cnvimass 5035 A preimage under any class...
cnvimarndm 5036 The preimage of the range ...
imasng 5037 The image of a singleton. ...
relimasn 5038 The image of a singleton. ...
elrelimasn 5039 Elementhood in the image o...
elimasn 5040 Membership in an image of ...
elimasng 5041 Membership in an image of ...
elimasni 5042 Membership in an image of ...
args 5043 Two ways to express the cl...
eliniseg 5044 Membership in an initial s...
epini 5045 Any set is equal to its pr...
iniseg 5046 An idiom that signifies an...
dffr3 5047 Alternate definition of we...
dfse2 5048 Alternate definition of se...
exse2 5049 Any set relation is set-li...
imass1 5050 Subset theorem for image. ...
imass2 5051 Subset theorem for image. ...
ndmima 5052 The image of a singleton o...
relcnv 5053 A converse is a relation. ...
relbrcnvg 5054 When ` R ` is a relation, ...
eliniseg2 5055 Eliminate the class existe...
relbrcnv 5056 When ` R ` is a relation, ...
cotr 5057 Two ways of saying a relat...
issref 5058 Two ways to state a relati...
cnvsym 5059 Two ways of saying a relat...
intasym 5060 Two ways of saying a relat...
asymref 5061 Two ways of saying a relat...
asymref2 5062 Two ways of saying a relat...
intirr 5063 Two ways of saying a relat...
brcodir 5064 Two ways of saying that tw...
codir 5065 Two ways of saying a relat...
qfto 5066 A quantifier-free way of e...
xpidtr 5067 A square cross product ` (...
trin2 5068 The intersection of two tr...
poirr2 5069 A partial order relation i...
trinxp 5070 The relation induced by a ...
soirri 5071 A strict order relation is...
sotri 5072 A strict order relation is...
son2lpi 5073 A strict order relation ha...
sotri2 5074 A transitivity relation. ...
sotri3 5075 A transitivity relation. ...
soirriOLD 5076 A strict order relation is...
sotriOLD 5077 A strict order relation is...
son2lpiOLD 5078 A strict order relation ha...
poleloe 5079 Express "less than or equa...
poltletr 5080 Transitive law for general...
somin1 5081 Property of a minimum in a...
somincom 5082 Commutativity of minimum i...
somin2 5083 Property of a minimum in a...
soltmin 5084 Being less than a minimum,...
cnvopab 5085 The converse of a class ab...
cnv0 5086 The converse of the empty ...
cnvi 5087 The converse of the identi...
cnvun 5088 The converse of a union is...
cnvdif 5089 Distributive law for conve...
cnvin 5090 Distributive law for conve...
rnun 5091 Distributive law for range...
rnin 5092 The range of an intersecti...
rniun 5093 The range of an indexed un...
rnuni 5094 The range of a union. Par...
imaundi 5095 Distributive law for image...
imaundir 5096 The image of a union. (Co...
dminss 5097 An upper bound for interse...
imainss 5098 An upper bound for interse...
cnvxp 5099 The converse of a cross pr...
xp0 5100 The cross product with the...
xpnz 5101 The cross product of nonem...
xpeq0 5102 At least one member of an ...
xpdisj1 5103 Cross products with disjoi...
xpdisj2 5104 Cross products with disjoi...
xpsndisj 5105 Cross products with two di...
djudisj 5106 Disjoint unions with disjo...
resdisj 5107 A double restriction to di...
rnxp 5108 The range of a cross produ...
dmxpss 5109 The domain of a cross prod...
rnxpss 5110 The range of a cross produ...
rnxpid 5111 The range of a square cros...
ssxpb 5112 A cross-product subclass r...
xp11 5113 The cross product of non-e...
xpcan 5114 Cancellation law for cross...
xpcan2 5115 Cancellation law for cross...
xpexr 5116 If a cross product is a se...
xpexr2 5117 If a nonempty cross produc...
ssrnres 5118 Subset of the range of a r...
rninxp 5119 Range of the intersection ...
dminxp 5120 Domain of the intersection...
imainrect 5121 Image of a relation restri...
sossfld 5122 The base set of a strict o...
sofld 5123 The base set of a nonempty...
soex 5124 If the relation in a stric...
cnvcnv3 5125 The set of all ordered pai...
dfrel2 5126 Alternate definition of re...
dfrel4v 5127 A relation can be expresse...
cnvcnv 5128 The double converse of a c...
cnvcnv2 5129 The double converse of a c...
cnvcnvss 5130 The double converse of a c...
cnveqb 5131 Equality theorem for conve...
cnveq0 5132 A relation empty iff its c...
dfrel3 5133 Alternate definition of re...
dmresv 5134 The domain of a universal ...
rnresv 5135 The range of a universal r...
dfrn4 5136 Range defined in terms of ...
rescnvcnv 5137 The restriction of the dou...
cnvcnvres 5138 The double converse of the...
imacnvcnv 5139 The image of the double co...
dmsnn0 5140 The domain of a singleton ...
rnsnn0 5141 The range of a singleton i...
dmsn0 5142 The domain of the singleto...
cnvsn0 5143 The converse of the single...
dmsn0el 5144 The domain of a singleton ...
relsn2 5145 A singleton is a relation ...
dmsnopg 5146 The domain of a singleton ...
dmsnopss 5147 The domain of a singleton ...
dmpropg 5148 The domain of an unordered...
dmsnop 5149 The domain of a singleton ...
dmprop 5150 The domain of an unordered...
dmtpop 5151 The domain of an unordered...
cnvcnvsn 5152 Double converse of a singl...
dmsnsnsn 5153 The domain of the singleto...
rnsnopg 5154 The range of a singleton o...
rnsnop 5155 The range of a singleton o...
op1sta 5156 Extract the first member o...
cnvsn 5157 Converse of a singleton of...
op2ndb 5158 Extract the second member ...
op2nda 5159 Extract the second member ...
cnvsng 5160 Converse of a singleton of...
opswap 5161 Swap the members of an ord...
elxp4 5162 Membership in a cross prod...
elxp5 5163 Membership in a cross prod...
cnvresima 5164 An image under the convers...
resdm2 5165 A class restricted to its ...
resdmres 5166 Restriction to the domain ...
imadmres 5167 The image of the domain of...
mptpreima 5168 The preimage of a function...
mptiniseg 5169 Converse singleton image o...
dmmpt 5170 The domain of the mapping ...
dmmptss 5171 The domain of a mapping is...
dmmptg 5172 The domain of the mapping ...
relco 5173 A composition is a relatio...
dfco2 5174 Alternate definition of a ...
dfco2a 5175 Generalization of ~ dfco2 ...
coundi 5176 Class composition distribu...
coundir 5177 Class composition distribu...
cores 5178 Restricted first member of...
resco 5179 Associative law for the re...
imaco 5180 Image of the composition o...
rnco 5181 The range of the compositi...
rnco2 5182 The range of the compositi...
dmco 5183 The domain of a compositio...
coiun 5184 Composition with an indexe...
cocnvcnv1 5185 A composition is not affec...
cocnvcnv2 5186 A composition is not affec...
cores2 5187 Absorption of a reverse (p...
co02 5188 Composition with the empty...
co01 5189 Composition with the empty...
coi1 5190 Composition with the ident...
coi2 5191 Composition with the ident...
coires1 5192 Composition with a restric...
coass 5193 Associative law for class ...
relcnvtr 5194 A relation is transitive i...
relssdmrn 5195 A relation is included in ...
cnvssrndm 5196 The converse is a subset o...
cossxp 5197 Composition as a subset of...
relrelss 5198 Two ways to describe the s...
unielrel 5199 The membership relation fo...
relfld 5200 The double union of a rela...
relresfld 5201 Restriction of a relation ...
relcoi2 5202 Composition with the ident...
relcoi1 5203 Composition with the ident...
unidmrn 5204 The double union of the co...
relcnvfld 5205 if ` R ` is a relation, it...
dfdm2 5206 Alternate definition of do...
unixp 5207 The double class union of ...
unixp0 5208 A cross product is empty i...
unixpid 5209 Field of a square cross pr...
cnvexg 5210 The converse of a set is a...
cnvex 5211 The converse of a set is a...
relcnvexb 5212 A relation is a set iff it...
ressn 5213 Restriction of a class to ...
cnviin 5214 The converse of an interse...
cnvpo 5215 The converse of a partial ...
cnvso 5216 The converse of a strict o...
coexg 5217 The composition of two set...
coex 5218 The composition of two set...
iotajust 5220 Soundness justification th...
dfiota2 5222 Alternate definition for d...
nfiota1 5223 Bound-variable hypothesis ...
nfiotad 5224 Deduction version of ~ nfi...
nfiota 5225 Bound-variable hypothesis ...
cbviota 5226 Change bound variables in ...
cbviotav 5227 Change bound variables in ...
sb8iota 5228 Variable substitution in d...
iotaeq 5229 Equality theorem for descr...
iotabi 5230 Equivalence theorem for de...
uniabio 5231 Part of Theorem 8.17 in [Q...
iotaval 5232 Theorem 8.19 in [Quine] p....
iotauni 5233 Equivalence between two di...
iotaint 5234 Equivalence between two di...
iota1 5235 Property of iota. (Contri...
iotanul 5236 Theorem 8.22 in [Quine] p....
iotassuni 5237 The ` iota ` class is a su...
iotaex 5238 Theorem 8.23 in [Quine] p....
iota4 5239 Theorem *14.22 in [Whitehe...
iota4an 5240 Theorem *14.23 in [Whitehe...
iota5 5241 A method for computing iot...
iotabidv 5242 Formula-building deduction...
iotabii 5243 Formula-building deduction...
iotacl 5244 Membership law for descrip...
iota2df 5245 A condition that allows us...
iota2d 5246 A condition that allows us...
iota2 5247 The unique element such th...
sniota 5248 A class abstraction with a...
dfiota4 5249 The ` iota ` operation usi...
csbiotag 5250 Class substitution within ...
dffun2 5267 Alternate definition of a ...
dffun3 5268 Alternate definition of fu...
dffun4 5269 Alternate definition of a ...
dffun5 5270 Alternate definition of fu...
dffun6f 5271 Definition of function, us...
dffun6 5272 Alternate definition of a ...
funmo 5273 A function has at most one...
funrel 5274 A function is a relation. ...
funss 5275 Subclass theorem for funct...
funeq 5276 Equality theorem for funct...
funeqi 5277 Equality inference for the...
funeqd 5278 Equality deduction for the...
nffun 5279 Bound-variable hypothesis ...
funeu 5280 There is exactly one value...
funeu2 5281 There is exactly one value...
dffun7 5282 Alternate definition of a ...
dffun8 5283 Alternate definition of a ...
dffun9 5284 Alternate definition of a ...
funfn 5285 An equivalence for the fun...
funi 5286 The identity relation is a...
nfunv 5287 The universe is not a func...
funopg 5288 A Kuratowski ordered pair ...
funopab 5289 A class of ordered pairs i...
funopabeq 5290 A class of ordered pairs o...
funopab4 5291 A class of ordered pairs o...
funmpt 5292 A function in maps-to nota...
funmpt2 5293 Functionality of a class g...
funco 5294 The composition of two fun...
funres 5295 A restriction of a functio...
funssres 5296 The restriction of a funct...
fun2ssres 5297 Equality of restrictions o...
funun 5298 The union of functions wit...
funcnvsn 5299 The converse singleton of ...
funsng 5300 A singleton of an ordered ...
fnsng 5301 Functionality and domain o...
funsn 5302 A singleton of an ordered ...
funprg 5303 A set of two pairs is a fu...
funpr 5304 A function with a domain o...
funtp 5305 A function with a domain o...
fnsn 5306 Functionality and domain o...
fnprg 5307 Domain of a function with ...
fntp 5308 A function with a domain o...
fun0 5309 The empty set is a functio...
funcnvcnv 5310 The double converse of a f...
funcnv2 5311 A simpler equivalence for ...
funcnv 5312 The converse of a class is...
funcnv3 5313 A condition showing a clas...
fun2cnv 5314 The double converse of a c...
svrelfun 5315 A single-valued relation i...
fncnv 5316 Single-rootedness (see ~ f...
fun11 5317 Two ways of stating that `...
fununi 5318 The union of a chain (with...
funcnvuni 5319 The union of a chain (with...
fun11uni 5320 The union of a chain (with...
funin 5321 The intersection with a fu...
funres11 5322 The restriction of a one-t...
funcnvres 5323 The converse of a restrict...
cnvresid 5324 Converse of a restricted i...
funcnvres2 5325 The converse of a restrict...
funimacnv 5326 The image of the preimage ...
funimass1 5327 A kind of contraposition l...
funimass2 5328 A kind of contraposition l...
imadif 5329 The image of a difference ...
imain 5330 The image of an intersecti...
funimaexg 5331 Axiom of Replacement using...
funimaex 5332 The image of a set under a...
isarep1 5333 Part of a study of the Axi...
isarep2 5334 Part of a study of the Axi...
fneq1 5335 Equality theorem for funct...
fneq2 5336 Equality theorem for funct...
fneq1d 5337 Equality deduction for fun...
fneq2d 5338 Equality deduction for fun...
fneq12d 5339 Equality deduction for fun...
fneq1i 5340 Equality inference for fun...
fneq2i 5341 Equality inference for fun...
nffn 5342 Bound-variable hypothesis ...
fnfun 5343 A function with domain is ...
fnrel 5344 A function with domain is ...
fndm 5345 The domain of a function. ...
funfni 5346 Inference to convert a fun...
fndmu 5347 A function has a unique do...
fnbr 5348 The first argument of bina...
fnop 5349 The first argument of an o...
fneu 5350 There is exactly one value...
fneu2 5351 There is exactly one value...
fnun 5352 The union of two functions...
fnunsn 5353 Extension of a function wi...
fnco 5354 Composition of two functio...
fnresdm 5355 A function does not change...
fnresdisj 5356 A function restricted to a...
2elresin 5357 Membership in two function...
fnssresb 5358 Restriction of a function ...
fnssres 5359 Restriction of a function ...
fnresin1 5360 Restriction of a function'...
fnresin2 5361 Restriction of a function'...
fnres 5362 An equivalence for functio...
fnresi 5363 Functionality and domain o...
fnima 5364 The image of a function's ...
fn0 5365 A function with empty doma...
fnimadisj 5366 A class that is disjoint w...
fnimaeq0 5367 Images under a function ne...
dfmpt3 5368 Alternate definition for t...
fnopabg 5369 Functionality and domain o...
fnopab 5370 Functionality and domain o...
mptfng 5371 The maps-to notation defin...
fnmpt 5372 The maps-to notation defin...
mpt0 5373 A mapping operation with e...
fnmpti 5374 Functionality and domain o...
dmmpti 5375 Domain of an ordered-pair ...
mptun 5376 Union of mappings which ar...
feq1 5377 Equality theorem for funct...
feq2 5378 Equality theorem for funct...
feq3 5379 Equality theorem for funct...
feq23 5380 Equality theorem for funct...
feq1d 5381 Equality deduction for fun...
feq2d 5382 Equality deduction for fun...
feq12d 5383 Equality deduction for fun...
feq123d 5384 Equality deduction for fun...
feq1i 5385 Equality inference for fun...
feq2i 5386 Equality inference for fun...
feq23i 5387 Equality inference for fun...
feq23d 5388 Equality deduction for fun...
nff 5389 Bound-variable hypothesis ...
elimf 5390 Eliminate a mapping hypoth...
ffn 5391 A mapping is a function. ...
dffn2 5392 Any function is a mapping ...
ffun 5393 A mapping is a function. ...
frel 5394 A mapping is a relation. ...
fdm 5395 The domain of a mapping. ...
fdmi 5396 The domain of a mapping. ...
frn 5397 The range of a mapping. (...
dffn3 5398 A function maps to its ran...
fss 5399 Expanding the codomain of ...
fco 5400 Composition of two mapping...
fco2 5401 Functionality of a composi...
fssxp 5402 A mapping is a class of or...
fex2 5403 A function with bounded do...
funssxp 5404 Two ways of specifying a p...
ffdm 5405 A mapping is a partial fun...
opelf 5406 The members of an ordered ...
fun 5407 The union of two functions...
fun2 5408 The union of two functions...
fnfco 5409 Composition of two functio...
fssres 5410 Restriction of a function ...
fssres2 5411 Restriction of a restricte...
fresin 5412 An identity for the mappin...
resasplit 5413 If two functions agree on ...
fresaun 5414 The union of two functions...
fresaunres2 5415 From the union of two func...
fresaunres1 5416 From the union of two func...
fcoi1 5417 Composition of a mapping a...
fcoi2 5418 Composition of restricted ...
feu 5419 There is exactly one value...
fcnvres 5420 The converse of a restrict...
fimacnvdisj 5421 The preimage of a class di...
fint 5422 Function into an intersect...
fin 5423 Mapping into an intersecti...
fabexg 5424 Existence of a set of func...
fabex 5425 Existence of a set of func...
dmfex 5426 If a mapping is a set, its...
f0 5427 The empty function. (Cont...
f00 5428 A class is a function with...
fconst 5429 A cross product with a sin...
fconstg 5430 A cross product with a sin...
fnconstg 5431 A cross product with a sin...
fconst6g 5432 Constant function with loo...
fconst6 5433 A constant function as a m...
f1eq1 5434 Equality theorem for one-t...
f1eq2 5435 Equality theorem for one-t...
f1eq3 5436 Equality theorem for one-t...
nff1 5437 Bound-variable hypothesis ...
dff12 5438 Alternate definition of a ...
f1f 5439 A one-to-one mapping is a ...
f1fn 5440 A one-to-one mapping is a ...
f1fun 5441 A one-to-one mapping is a ...
f1rel 5442 A one-to-one onto mapping ...
f1dm 5443 The domain of a one-to-one...
f1ss 5444 A function that is one-to-...
f1ssr 5445 Combine a one-to-one funct...
f1ssres 5446 A function that is one-to-...
f1cnvcnv 5447 Two ways to express that a...
f1co 5448 Composition of one-to-one ...
foeq1 5449 Equality theorem for onto ...
foeq2 5450 Equality theorem for onto ...
foeq3 5451 Equality theorem for onto ...
nffo 5452 Bound-variable hypothesis ...
fof 5453 An onto mapping is a mappi...
fofun 5454 An onto mapping is a funct...
fofn 5455 An onto mapping is a funct...
forn 5456 The codomain of an onto fu...
dffo2 5457 Alternate definition of an...
foima 5458 The image of the domain of...
dffn4 5459 A function maps onto its r...
funforn 5460 A function maps its domain...
fodmrnu 5461 An onto function has uniqu...
fores 5462 Restriction of a function....
foco 5463 Composition of onto functi...
foconst 5464 A nonzero constant functio...
f1oeq1 5465 Equality theorem for one-t...
f1oeq2 5466 Equality theorem for one-t...
f1oeq3 5467 Equality theorem for one-t...
f1oeq23 5468 Equality theorem for one-t...
f1eq123d 5469 Equality deduction for one...
foeq123d 5470 Equality deduction for ont...
f1oeq123d 5471 Equality deduction for one...
nff1o 5472 Bound-variable hypothesis ...
f1of1 5473 A one-to-one onto mapping ...
f1of 5474 A one-to-one onto mapping ...
f1ofn 5475 A one-to-one onto mapping ...
f1ofun 5476 A one-to-one onto mapping ...
f1orel 5477 A one-to-one onto mapping ...
f1odm 5478 The domain of a one-to-one...
dff1o2 5479 Alternate definition of on...
dff1o3 5480 Alternate definition of on...
f1ofo 5481 A one-to-one onto function...
dff1o4 5482 Alternate definition of on...
dff1o5 5483 Alternate definition of on...
f1orn 5484 A one-to-one function maps...
f1f1orn 5485 A one-to-one function maps...
f1oabexg 5486 The class of all 1-1-onto ...
f1ocnv 5487 The converse of a one-to-o...
f1ocnvb 5488 A relation is a one-to-one...
f1ores 5489 The restriction of a one-t...
f1orescnv 5490 The converse of a one-to-o...
f1imacnv 5491 Preimage of an image. (Co...
foimacnv 5492 A reverse version of ~ f1i...
foun 5493 The union of two onto func...
f1oun 5494 The union of two one-to-on...
fun11iun 5495 The union of a chain (with...
resdif 5496 The restriction of a one-t...
resin 5497 The restriction of a one-t...
f1oco 5498 Composition of one-to-one ...
f1cnv 5499 The converse of an injecti...
funcocnv2 5500 Composition with the conve...
fococnv2 5501 The composition of an onto...
f1ococnv2 5502 The composition of a one-t...
f1cocnv2 5503 Composition of an injectiv...
f1ococnv1 5504 The composition of a one-t...
f1cocnv1 5505 Composition of an injectiv...
funcoeqres 5506 Re-express a constraint on...
ffoss 5507 Relationship between a map...
f11o 5508 Relationship between one-t...
f10 5509 The empty set maps one-to-...
f1o00 5510 One-to-one onto mapping of...
fo00 5511 Onto mapping of the empty ...
f1o0 5512 One-to-one onto mapping of...
f1oi 5513 A restriction of the ident...
f1ovi 5514 The identity relation is a...
f1osn 5515 A singleton of an ordered ...
f1osng 5516 A singleton of an ordered ...
f1oprswap 5517 A two-element swap is a bi...
tz6.12-2 5518 Function value when ` F ` ...
fveu 5519 The value of a function at...
brprcneu 5520 If ` A ` is a proper class...
fvprc 5521 A function's value at a pr...
fv2 5522 Alternate definition of fu...
dffv3 5523 A definition of function v...
dffv4 5524 The previous definition of...
elfv 5525 Membership in a function v...
fveq1 5526 Equality theorem for funct...
fveq2 5527 Equality theorem for funct...
fveq1i 5528 Equality inference for fun...
fveq1d 5529 Equality deduction for fun...
fveq2i 5530 Equality inference for fun...
fveq2d 5531 Equality deduction for fun...
fveq12i 5532 Equality deduction for fun...
fveq12d 5533 Equality deduction for fun...
nffv 5534 Bound-variable hypothesis ...
nffvmpt1 5535 Bound-variable hypothesis ...
nffvd 5536 Deduction version of bound...
csbfv12g 5537 Move class substitution in...
csbfv12gALT 5538 Move class substitution in...
csbfv2g 5539 Move class substitution in...
csbfvg 5540 Substitution for a functio...
fvex 5541 The value of a class exist...
fvif 5542 Move a conditional outside...
fv3 5543 Alternate definition of th...
fvres 5544 The value of a restricted ...
funssfv 5545 The value of a member of t...
tz6.12-1 5546 Function value. Theorem 6...
tz6.12 5547 Function value. Theorem 6...
tz6.12f 5548 Function value, using boun...
tz6.12c 5549 Corollary of Theorem 6.12(...
tz6.12i 5550 Corollary of Theorem 6.12(...
fvbr0 5551 Two possibilities for the ...
fvrn0 5552 A function value is a memb...
fvssunirn 5553 The result of a function v...
ndmfv 5554 The value of a class outsi...
ndmfvrcl 5555 Reverse closure law for fu...
elfvdm 5556 If a function value has a ...
elfvex 5557 If a function value has a ...
elfvexd 5558 If a function value is non...
nfvres 5559 The value of a non-member ...
nfunsn 5560 If the restriction of a cl...
fv01 5561 Function value of the empt...
fveqres 5562 Equal values imply equal v...
funbrfv 5563 The second argument of a b...
funopfv 5564 The second element in an o...
fnbrfvb 5565 Equivalence of function va...
fnopfvb 5566 Equivalence of function va...
funbrfvb 5567 Equivalence of function va...
funopfvb 5568 Equivalence of function va...
funbrfv2b 5569 Function value in terms of...
dffn5 5570 Representation of a functi...
fnrnfv 5571 The range of a function ex...
fvelrnb 5572 A member of a function's r...
dfimafn 5573 Alternate definition of th...
dfimafn2 5574 Alternate definition of th...
funimass4 5575 Membership relation for th...
fvelima 5576 Function value in an image...
feqmptd 5577 Deduction form of ~ dffn5 ...
feqresmpt 5578 Express a restricted funct...
dffn5f 5579 Representation of a functi...
fvelimab 5580 Function value in an image...
fvi 5581 The value of the identity ...
fviss 5582 The value of the identity ...
fniinfv 5583 The indexed intersection o...
fnsnfv 5584 Singleton of function valu...
fnimapr 5585 The image of a pair under ...
ssimaex 5586 The existence of a subimag...
ssimaexg 5587 The existence of a subimag...
funfv 5588 A simplified expression fo...
funfv2 5589 The value of a function. ...
funfv2f 5590 The value of a function. ...
fvun 5591 Value of the union of two ...
fvun1 5592 The value of a union when ...
fvun2 5593 The value of a union when ...
dffv2 5594 Alternate definition of fu...
dmfco 5595 Domains of a function comp...
fvco2 5596 Value of a function compos...
fvco 5597 Value of a function compos...
fvco3 5598 Value of a function compos...
fvco4i 5599 Conditions for a compositi...
fvopab3g 5600 Value of a function given ...
fvopab3ig 5601 Value of a function given ...
fvmptg 5602 Value of a function given ...
fvmpti 5603 Value of a function given ...
fvmpt 5604 Value of a function given ...
fvmpts 5605 Value of a function given ...
fvmpt3 5606 Value of a function given ...
fvmpt3i 5607 Value of a function given ...
fvmptd 5608 Deduction version of ~ fvm...
fvmpt2i 5609 Value of a function given ...
fvmpt2 5610 Value of a function given ...
fvmptss 5611 If all the values of the m...
fvmptex 5612 Express a function ` F ` w...
fvmptdf 5613 Alternate deduction versio...
fvmptdv 5614 Alternate deduction versio...
fvmptdv2 5615 Alternate deduction versio...
mpteqb 5616 Bidirectional equality the...
fvmptt 5617 Closed theorem form of ~ f...
fvmptf 5618 Value of a function given ...
fvmptnf 5619 The value of a function gi...
fvmptn 5620 This somewhat non-intuitiv...
fvmptss2 5621 A mapping always evaluates...
fvopab4ndm 5622 Value of a function given ...
fvopab6 5623 Value of a function given ...
eqfnfv 5624 Equality of functions is d...
eqfnfv2 5625 Equality of functions is d...
eqfnfv3 5626 Derive equality of functio...
eqfnfvd 5627 Deduction for equality of ...
eqfnfv2f 5628 Equality of functions is d...
eqfunfv 5629 Equality of functions is d...
fvreseq 5630 Equality of restricted fun...
fndmdif 5631 Two ways to express the lo...
fndmdifcom 5632 The difference set between...
fndmdifeq0 5633 The difference set of two ...
fndmin 5634 Two ways to express the lo...
fneqeql 5635 Two functions are equal if...
fneqeql2 5636 Two functions are equal if...
fnreseql 5637 Two functions are equal on...
chfnrn 5638 The range of a choice func...
funfvop 5639 Ordered pair with function...
funfvbrb 5640 Two ways to say that ` A `...
fvimacnvi 5641 A member of a preimage is ...
fvimacnv 5642 The argument of a function...
funimass3 5643 A kind of contraposition l...
funimass5 5644 A subclass of a preimage i...
funconstss 5645 Two ways of specifying tha...
fvimacnvALT 5646 Another proof of ~ fvimacn...
elpreima 5647 Membership in the preimage...
fniniseg 5648 Membership in the preimage...
fncnvima2 5649 Inverse images under funct...
fniniseg2 5650 Inverse point images under...
fnniniseg2 5651 Support sets of functions ...
rexsupp 5652 Existential quantification...
unpreima 5653 Preimage of a union. (Con...
inpreima 5654 Preimage of an intersectio...
difpreima 5655 Preimage of a difference. ...
respreima 5656 The preimage of a restrict...
iinpreima 5657 Preimage of an intersectio...
intpreima 5658 Preimage of an intersectio...
fimacnv 5659 The preimage of the codoma...
suppss 5660 Show that the support of a...
suppssr 5661 A function is zero outside...
fnopfv 5662 Ordered pair with function...
fvelrn 5663 A function's value belongs...
fnfvelrn 5664 A function's value belongs...
ffvelrn 5665 A function's value belongs...
ffvelrni 5666 A function's value belongs...
ffvelrnda 5667 A function's value belongs...
ffvelrnd 5668 A function's value belongs...
rexrn 5669 Restricted existential qua...
ralrn 5670 Restricted universal quant...
ralrnmpt 5671 A restricted quantifier ov...
rexrnmpt 5672 A restricted quantifier ov...
f0cli 5673 Unconditional closure of a...
dff2 5674 Alternate definition of a ...
dff3 5675 Alternate definition of a ...
dff4 5676 Alternate definition of a ...
dffo3 5677 An onto mapping expressed ...
dffo4 5678 Alternate definition of an...
dffo5 5679 Alternate definition of an...
exfo 5680 A relation equivalent to t...
foelrn 5681 Property of a surjective f...
foco2 5682 If a composition of two fu...
fmpt 5683 Functionality of the mappi...
f1ompt 5684 Express bijection for a ma...
fmpti 5685 Functionality of the mappi...
fmptd 5686 Domain and codomain of the...
ffnfv 5687 A function maps to a class...
ffnfvf 5688 A function maps to a class...
fnfvrnss 5689 An upper bound for range d...
fmpt2d 5690 Domain and codomain of the...
fmpt2dOLD 5691 Domain and codomain of the...
ffvresb 5692 A necessary and sufficient...
fmptco 5693 Composition of two functio...
fmptcof 5694 Version of ~ fmptco where ...
fmptcos 5695 Composition of two functio...
fcompt 5696 Express composition of two...
fcoconst 5697 Composition with a constan...
fsn 5698 A function maps a singleto...
fsng 5699 A function maps a singleto...
fsn2 5700 A function that maps a sin...
xpsng 5701 The cross product of two s...
xpsn 5702 The cross product of two s...
dfmpt 5703 Alternate definition for t...
fnasrn 5704 A function expressed as th...
ressnop0 5705 If ` A ` is not in ` C ` ,...
fpr 5706 A function with a domain o...
fnressn 5707 A function restricted to a...
funressn 5708 A function restricted to a...
fressnfv 5709 The value of a function re...
fvconst 5710 The value of a constant fu...
fmptsn 5711 Express a singleton functi...
fmptap 5712 Append an additional value...
fvresi 5713 The value of a restricted ...
fvunsn 5714 Remove an ordered pair not...
fvsn 5715 The value of a singleton o...
fvsng 5716 The value of a singleton o...
fvsnun1 5717 The value of a function wi...
fvsnun2 5718 The value of a function wi...
fnsnsplit 5719 Split a function into a si...
fsnunf 5720 Adjoining a point to a fun...
fsnunf2 5721 Adjoining a point to a pun...
fsnunfv 5722 Recover the added point fr...
fsnunres 5723 Recover the original funct...
fvpr1 5724 The value of a function wi...
fvpr2 5725 The value of a function wi...
fvtp1 5726 The first value of a funct...
fvtp2 5727 The second value of a func...
fvtp3 5728 The third value of a funct...
fvconst2g 5729 The value of a constant fu...
fconst2g 5730 A constant function expres...
fvconst2 5731 The value of a constant fu...
fconst2 5732 A constant function expres...
fconst5 5733 Two ways to express that a...
fnsuppres 5734 Two ways to express restri...
fnsuppeq0 5735 The support of a function ...
fconstfv 5736 A constant function expres...
fconst3 5737 Two ways to express a cons...
fconst4 5738 Two ways to express a cons...
resfunexg 5739 The restriction of a funct...
resfunexgALT 5740 The restriction of a funct...
cofunexg 5741 Existence of a composition...
cofunex2g 5742 Existence of a composition...
fnex 5743 If the domain of a functio...
fnexALT 5744 If the domain of a functio...
funex 5745 If the domain of a functio...
opabex 5746 Existence of a function ex...
mptexg 5747 If the domain of a functio...
mptex 5748 If the domain of a functio...
funrnex 5749 If the domain of a functio...
zfrep6 5750 A version of the Axiom of ...
fex 5751 If the domain of a mapping...
fornex 5752 If the domain of an onto f...
f1dmex 5753 If the codomain of a one-t...
eufnfv 5754 A function is uniquely det...
funfvima 5755 A function's value in a pr...
funfvima2 5756 A function's value in an i...
funfvima3 5757 A class including a functi...
fnfvima 5758 The function value of an o...
rexima 5759 Existential quantification...
ralima 5760 Universal quantification u...
idref 5761 TODO: This is the same as...
fvclss 5762 Upper bound for the class ...
fvclex 5763 Existence of the class of ...
fvresex 5764 Existence of the class of ...
abrexex 5765 Existence of a class abstr...
abrexexg 5766 Existence of a class abstr...
elabrex 5767 Elementhood in an image se...
abrexco 5768 Composition of two image m...
iunexg 5769 The existence of an indexe...
abrexex2g 5770 Existence of an existentia...
opabex3 5771 Existence of an ordered pa...
iunex 5772 The existence of an indexe...
imaiun 5773 The image of an indexed un...
imauni 5774 The image of a union is th...
fniunfv 5775 The indexed union of a fun...
funiunfv 5776 The indexed union of a fun...
funiunfvf 5777 The indexed union of a fun...
eluniima 5778 Membership in the union of...
elunirn 5779 Membership in the union of...
fnunirn 5780 Membership in a union of s...
elunirnALT 5781 Membership in the union of...
abrexex2 5782 Existence of an existentia...
abexssex 5783 Existence of a class abstr...
abexex 5784 A condition where a class ...
dff13 5785 A one-to-one function in t...
dff13f 5786 A one-to-one function in t...
f1mpt 5787 Express injection for a ma...
f1fveq 5788 Equality of function value...
f1elima 5789 Membership in the image of...
f1imass 5790 Taking images under a one-...
f1imaeq 5791 Taking images under a one-...
f1imapss 5792 Taking images under a one-...
dff1o6 5793 A one-to-one onto function...
f1ocnvfv1 5794 The converse value of the ...
f1ocnvfv2 5795 The value of the converse ...
f1ocnvfv 5796 Relationship between the v...
f1ocnvfvb 5797 Relationship between the v...
f1ocnvdm 5798 The value of the converse ...
fcof1 5799 An application is injectiv...
fcofo 5800 An application is surjecti...
cbvfo 5801 Change bound variable betw...
cbvexfo 5802 Change bound variable betw...
cocan1 5803 An injection is left-cance...
cocan2 5804 A surjection is right-canc...
fcof1o 5805 Show that two functions ar...
foeqcnvco 5806 Condition for function equ...
f1eqcocnv 5807 Condition for function equ...
fveqf1o 5808 Given a bijection ` F ` , ...
fliftrel 5809 ` F ` , a function lift, i...
fliftel 5810 Elementhood in the relatio...
fliftel1 5811 Elementhood in the relatio...
fliftcnv 5812 Converse of the relation `...
fliftfun 5813 The function ` F ` is the ...
fliftfund 5814 The function ` F ` is the ...
fliftfuns 5815 The function ` F ` is the ...
fliftf 5816 The domain and range of th...
fliftval 5817 The value of the function ...
isoeq1 5818 Equality theorem for isomo...
isoeq2 5819 Equality theorem for isomo...
isoeq3 5820 Equality theorem for isomo...
isoeq4 5821 Equality theorem for isomo...
isoeq5 5822 Equality theorem for isomo...
nfiso 5823 Bound-variable hypothesis ...
isof1o 5824 An isomorphism is a one-to...
isorel 5825 An isomorphism connects bi...
soisores 5826 Express the condition of i...
soisoi 5827 Infer isomorphism from one...
isoid 5828 Identity law for isomorphi...
isocnv 5829 Converse law for isomorphi...
isocnv2 5830 Converse law for isomorphi...
isocnv3 5831 Complementation law for is...
isores2 5832 An isomorphism from one we...
isores1 5833 An isomorphism from one we...
isores3 5834 Induced isomorphism on a s...
isotr 5835 Composition (transitive) l...
isomin 5836 Isomorphisms preserve mini...
isoini 5837 Isomorphisms preserve init...
isoini2 5838 Isomorphisms are isomorphi...
isofrlem 5839 Lemma for ~ isofr . (Cont...
isoselem 5840 Lemma for ~ isose . (Cont...
isofr 5841 An isomorphism preserves w...
isose 5842 An isomorphism preserves s...
isofr2 5843 A weak form of ~ isofr tha...
isopolem 5844 Lemma for ~ isopo . (Cont...
isopo 5845 An isomorphism preserves p...
isosolem 5846 Lemma for ~ isoso . (Cont...
isoso 5847 An isomorphism preserves s...
isowe 5848 An isomorphism preserves w...
isowe2 5849 A weak form of ~ isowe tha...
f1oiso 5850 Any one-to-one onto functi...
f1oiso2 5851 Any one-to-one onto functi...
f1owe 5852 Well-ordering of isomorphi...
f1oweALT 5853 Well-ordering of isomorphi...
weniso 5854 A set-like well-ordering h...
weisoeq 5855 Thus, there is at most one...
weisoeq2 5856 Thus, there is at most one...
wemoiso 5857 Thus, there is at most one...
wemoiso2 5858 Thus, there is at most one...
knatar 5859 The Knaster-Tarski theorem...
oveq 5866 Equality theorem for opera...
oveq1 5867 Equality theorem for opera...
oveq2 5868 Equality theorem for opera...
oveq12 5869 Equality theorem for opera...
oveq1i 5870 Equality inference for ope...
oveq2i 5871 Equality inference for ope...
oveq12i 5872 Equality inference for ope...
oveqi 5873 Equality inference for ope...
oveq123i 5874 Equality inference for ope...
oveq1d 5875 Equality deduction for ope...
oveq2d 5876 Equality deduction for ope...
oveqd 5877 Equality deduction for ope...
oveq12d 5878 Equality deduction for ope...
oveqan12d 5879 Equality deduction for ope...
oveqan12rd 5880 Equality deduction for ope...
oveq123d 5881 Equality deduction for ope...
nfovd 5882 Deduction version of bound...
nfov 5883 Bound-variable hypothesis ...
oprabid 5884 The law of concretion. Sp...
ovex 5885 The result of an operation...
ovssunirn 5886 The result of an operation...
ovprc 5887 The value of an operation ...
ovprc1 5888 The value of an operation ...
ovprc2 5889 The value of an operation ...
ovrcl 5890 Reverse closure for an ope...
csbovg 5891 Move class substitution in...
csbov12g 5892 Move class substitution in...
csbov1g 5893 Move class substitution in...
csbov2g 5894 Move class substitution in...
rspceov 5895 A frequently used special ...
fnotovb 5896 Equivalence of operation v...
dfoprab2 5897 Class abstraction for oper...
reloprab 5898 An operation class abstrac...
nfoprab1 5899 The abstraction variables ...
nfoprab2 5900 The abstraction variables ...
nfoprab3 5901 The abstraction variables ...
nfoprab 5902 Bound-variable hypothesis ...
oprabbid 5903 Equivalent wff's yield equ...
oprabbidv 5904 Equivalent wff's yield equ...
oprabbii 5905 Equivalent wff's yield equ...
ssoprab2 5906 Equivalence of ordered pai...
ssoprab2b 5907 Equivalence of ordered pai...
eqoprab2b 5908 Equivalence of ordered pai...
mpt2eq123 5909 An equality theorem for th...
mpt2eq12 5910 An equality theorem for th...
mpt2eq123dva 5911 An equality deduction for ...
mpt2eq123dv 5912 An equality deduction for ...
mpt2eq123i 5913 An equality inference for ...
mpt2eq3dva 5914 Slightly more general equa...
mpt2eq3ia 5915 An equality inference for ...
nfmpt21 5916 Bound-variable hypothesis ...
nfmpt22 5917 Bound-variable hypothesis ...
nfmpt2 5918 Bound-variable hypothesis ...
oprab4 5919 Two ways to state the doma...
cbvoprab1 5920 Rule used to change first ...
cbvoprab2 5921 Change the second bound va...
cbvoprab12 5922 Rule used to change first ...
cbvoprab12v 5923 Rule used to change first ...
cbvoprab3 5924 Rule used to change the th...
cbvoprab3v 5925 Rule used to change the th...
cbvmpt2x 5926 Rule to change the bound v...
cbvmpt2 5927 Rule to change the bound v...
cbvmpt2v 5928 Rule to change the bound v...
elimdelov 5929 Eliminate a hypothesis whi...
dmoprab 5930 The domain of an operation...
dmoprabss 5931 The domain of an operation...
rnoprab 5932 The range of an operation ...
rnoprab2 5933 The range of a restricted ...
reldmoprab 5934 The domain of an operation...
oprabss 5935 Structure of an operation ...
eloprabga 5936 The law of concretion for ...
eloprabg 5937 The law of concretion for ...
ssoprab2i 5938 Inference of operation cla...
mpt2v 5939 Operation with universal d...
mpt2mptx 5940 Express a two-argument fun...
mpt2mpt 5941 Express a two-argument fun...
resoprab 5942 Restriction of an operatio...
resoprab2 5943 Restriction of an operator...
resmpt2 5944 Restriction of the mapping...
funoprabg 5945 "At most one" is a suffici...
funoprab 5946 "At most one" is a suffici...
fnoprabg 5947 Functionality and domain o...
mpt2fun 5948 The maps-to notation for a...
fnoprab 5949 Functionality and domain o...
ffnov 5950 An operation maps to a cla...
fovcl 5951 Closure law for an operati...
eqfnov 5952 Equality of two operations...
eqfnov2 5953 Two operators with the sam...
fnov 5954 Representation of a functi...
mpt22eqb 5955 Bidirectional equality the...
rnmpt2 5956 The range of an operation ...
reldmmpt2 5957 The domain of an operation...
elrnmpt2g 5958 Membership in the range of...
elrnmpt2 5959 Membership in the range of...
ralrnmpt2 5960 A restricted quantifier ov...
rexrnmpt2 5961 A restricted quantifier ov...
oprabexd 5962 Existence of an operator a...
oprabex 5963 Existence of an operation ...
oprabex3 5964 Existence of an operation ...
oprabrexex2 5965 Existence of an existentia...
ovid 5966 The value of an operation ...
ovidig 5967 The value of an operation ...
ovidi 5968 The value of an operation ...
ov 5969 The value of an operation ...
ovigg 5970 The value of an operation ...
ovig 5971 The value of an operation ...
ovmpt4g 5972 Value of a function given ...
ovmpt2s 5973 Value of a function given ...
ov2gf 5974 The value of an operation ...
ovmpt2dxf 5975 Value of an operation give...
ovmpt2dx 5976 Value of an operation give...
ovmpt2d 5977 Value of an operation give...
ovmpt2x 5978 The value of an operation ...
ovmpt2ga 5979 Value of an operation give...
ovmpt2a 5980 Value of an operation give...
ovmpt2df 5981 Alternate deduction versio...
ovmpt2dv 5982 Alternate deduction versio...
ovmpt2dv2 5983 Alternate deduction versio...
ovmpt2g 5984 Value of an operation give...
ovmpt2 5985 Value of an operation give...
ov3 5986 The value of an operation ...
ov6g 5987 The value of an operation ...
ovg 5988 The value of an operation ...
ovres 5989 The value of a restricted ...
ovresd 5990 Lemma for converting metri...
oprssov 5991 The value of a member of t...
fovrn 5992 An operation's value belon...
fovrnda 5993 An operation's value belon...
fovrnd 5994 An operation's value belon...
fnrnov 5995 The range of an operation ...
foov 5996 An onto mapping of an oper...
fnovrn 5997 An operation's value belon...
ovelrn 5998 A member of an operation's...
funimassov 5999 Membership relation for th...
ovelimab 6000 Operation value in an imag...
ovconst2 6001 The value of a constant op...
ab2rexex 6002 Existence of a class abstr...
ab2rexex2 6003 Existence of an existentia...
oprssdm 6004 Domain of closure of an op...
ndmovg 6005 The value of an operation ...
ndmov 6006 The value of an operation ...
ndmovcl 6007 The closure of an operatio...
ndmovrcl 6008 Reverse closure law, when ...
ndmovcom 6009 Any operation is commutati...
ndmovass 6010 Any operation is associati...
ndmovdistr 6011 Any operation is distribut...
ndmovord 6012 Elimination of redundant a...
ndmovordi 6013 Elimination of redundant a...
caovclg 6014 Convert an operation closu...
caovcld 6015 Convert an operation closu...
caovcl 6016 Convert an operation closu...
caovcomg 6017 Convert an operation commu...
caovcomd 6018 Convert an operation commu...
caovcom 6019 Convert an operation commu...
caovassg 6020 Convert an operation assoc...
caovassd 6021 Convert an operation assoc...
caovass 6022 Convert an operation assoc...
caovcang 6023 Convert an operation cance...
caovcand 6024 Convert an operation cance...
caovcanrd 6025 Commute the arguments of a...
caovcan 6026 Convert an operation cance...
caovordig 6027 Convert an operation order...
caovordid 6028 Convert an operation order...
caovordg 6029 Convert an operation order...
caovordd 6030 Convert an operation order...
caovord2d 6031 Operation ordering law wit...
caovord3d 6032 Ordering law. (Contribute...
caovord 6033 Convert an operation order...
caovord2 6034 Operation ordering law wit...
caovord3 6035 Ordering law. (Contribute...
caovdig 6036 Convert an operation distr...
caovdid 6037 Convert an operation distr...
caovdir2d 6038 Convert an operation distr...
caovdirg 6039 Convert an operation rever...
caovdird 6040 Convert an operation distr...
caovdi 6041 Convert an operation distr...
caov32d 6042 Rearrange arguments in a c...
caov12d 6043 Rearrange arguments in a c...
caov31d 6044 Rearrange arguments in a c...
caov13d 6045 Rearrange arguments in a c...
caov4d 6046 Rearrange arguments in a c...
caov411d 6047 Rearrange arguments in a c...
caov42d 6048 Rearrange arguments in a c...
caov32 6049 Rearrange arguments in a c...
caov12 6050 Rearrange arguments in a c...
caov31 6051 Rearrange arguments in a c...
caov13 6052 Rearrange arguments in a c...
caov4 6053 Rearrange arguments in a c...
caov411 6054 Rearrange arguments in a c...
caov42 6055 Rearrange arguments in a c...
caovdir 6056 Reverse distributive law. ...
caovdilem 6057 Lemma used by real number ...
caovlem2 6058 Lemma used in real number ...
caovmo 6059 Uniqueness of inverse elem...
grprinvlem 6060 Lemma for ~ grprinvd . (C...
grprinvd 6061 Deduce right inverse from ...
grpridd 6062 Deduce right identity from...
elmpt2cl 6063 If a two-parameter class i...
elmpt2cl1 6064 If a two-parameter class i...
elmpt2cl2 6065 If a two-parameter class i...
elovmpt2 6066 Utility lemma for two-para...
relmptopab 6067 Any function to sets of or...
f1ocnvd 6068 Describe an implicit one-t...
f1od 6069 Describe an implicit one-t...
f1ocnv2d 6070 Describe an implicit one-t...
f1o2d 6071 Describe an implicit one-t...
xpexgALT 6072 The cross product of two s...
f1opw2 6073 A one-to-one mapping induc...
f1opw 6074 A one-to-one mapping induc...
suppss2 6075 Show that the support of a...
suppssfv 6076 Formula building theorem f...
suppssov1 6077 Formula building theorem f...
ofeq 6082 Equality theorem for funct...
ofreq 6083 Equality theorem for funct...
ofexg 6084 A function operation restr...
nfof 6085 Hypothesis builder for fun...
nfofr 6086 Hypothesis builder for fun...
offval 6087 Value of an operation appl...
ofrfval 6088 Value of a relation applie...
ofval 6089 Evaluate a function operat...
ofrval 6090 Exhibit a function relatio...
offn 6091 The function operation pro...
fnfvof 6092 Function value of a pointw...
offval3 6093 General value of ` ( F oF ...
offres 6094 Pointwise combination comm...
off 6095 The function operation pro...
ofres 6096 Restrict the operands of a...
offval2 6097 The function operation exp...
ofrfval2 6098 The function relation acti...
ofco 6099 The composition of a funct...
offveq 6100 Convert an identity of the...
offveqb 6101 Equivalent expressions for...
ofc1 6102 Left operation by a consta...
ofc2 6103 Right operation by a const...
ofc12 6104 Function operation on two ...
caofref 6105 Transfer a reflexive law t...
caofinvl 6106 Transfer a left inverse la...
caofid0l 6107 Transfer a left identity l...
caofid0r 6108 Transfer a right identity ...
caofid1 6109 Transfer a right absorptio...
caofid2 6110 Transfer a right absorptio...
caofcom 6111 Transfer a commutative law...
caofrss 6112 Transfer a relation subset...
caofass 6113 Transfer an associative la...
caoftrn 6114 Transfer a transitivity la...
caofdi 6115 Transfer a distributive la...
caofdir 6116 Transfer a reverse distrib...
caonncan 6117 Transfer ~ nncan -shaped l...
ofmres 6118 Equivalent expressions for...
ofmresval 6119 Value of a restriction of ...
ofmresex 6120 Existence of a restriction...
suppssof1 6121 Formula building theorem f...
1stval 6126 The value of the function ...
2ndval 6127 The value of the function ...
1st0 6128 The value of the first-mem...
2nd0 6129 The value of the second-me...
op1st 6130 Extract the first member o...
op2nd 6131 Extract the second member ...
op1std 6132 Extract the first member o...
op2ndd 6133 Extract the second member ...
op1stg 6134 Extract the first member o...
op2ndg 6135 Extract the second member ...
ot1stg 6136 Extract the first member o...
ot2ndg 6137 Extract the second member ...
ot3rdg 6138 Extract the third member o...
1stval2 6139 Alternate value of the fun...
2ndval2 6140 Alternate value of the fun...
fo1st 6141 The ` 1st ` function maps ...
fo2nd 6142 The ` 2nd ` function maps ...
f1stres 6143 Mapping of a restriction o...
f2ndres 6144 Mapping of a restriction o...
fo1stres 6145 Onto mapping of a restrict...
fo2ndres 6146 Onto mapping of a restrict...
1st2val 6147 Value of an alternate defi...
2nd2val 6148 Value of an alternate defi...
1stcof 6149 Composition of the first m...
2ndcof 6150 Composition of the first m...
xp1st 6151 Location of the first elem...
xp2nd 6152 Location of the second ele...
elxp6 6153 Membership in a cross prod...
elxp7 6154 Membership in a cross prod...
difxp 6155 Difference of Cartesian pr...
difxp1 6156 Difference law for cross p...
difxp2 6157 Difference law for cross p...
eqopi 6158 Equality with an ordered p...
xp2 6159 Representation of cross pr...
unielxp 6160 The membership relation fo...
1st2nd2 6161 Reconstruction of a member...
1st2ndb 6162 Reconstruction of an order...
xpopth 6163 An ordered pair theorem fo...
eqop 6164 Two ways to express equali...
eqop2 6165 Two ways to express equali...
op1steq 6166 Two ways of expressing tha...
2nd1st 6167 Swap the members of an ord...
1st2nd 6168 Reconstruction of a member...
1stdm 6169 The first ordered pair com...
2ndrn 6170 The second ordered pair co...
1st2ndbr 6171 Express an element of a re...
releldm2 6172 Two ways of expressing mem...
reldm 6173 An expression for the doma...
sbcopeq1a 6174 Equality theorem for subst...
csbopeq1a 6175 Equality theorem for subst...
dfopab2 6176 A way to define an ordered...
dfoprab3s 6177 A way to define an operati...
dfoprab3 6178 Operation class abstractio...
dfoprab4 6179 Operation class abstractio...
dfoprab4f 6180 Operation class abstractio...
dfxp3 6181 Define the cross product o...
copsex2gb 6182 Implicit substitution infe...
copsex2ga 6183 Implicit substitution infe...
elopaba 6184 Membership in an ordered p...
exopxfr 6185 Transfer ordered-pair exis...
exopxfr2 6186 Transfer ordered-pair exis...
elopabi 6187 A consequence of membershi...
eloprabi 6188 A consequence of membershi...
mpt2mptsx 6189 Express a two-argument fun...
mpt2mpts 6190 Express a two-argument fun...
dmmpt2ssx 6191 The domain of a mapping is...
fmpt2x 6192 Functionality, domain and ...
fmpt2 6193 Functionality, domain and ...
fnmpt2 6194 Functionality and domain o...
fnmpt2i 6195 Functionality and domain o...
dmmpt2 6196 Domain of a class given by...
mpt2exxg 6197 Existence of an operation ...
mpt2exg 6198 Existence of an operation ...
mpt2exga 6199 If the domain of a functio...
mpt2ex 6200 If the domain of a functio...
mpt20 6201 A mapping operation with e...
ovmptss 6202 If all the values of the m...
relmpt2opab 6203 Any function to sets of or...
fmpt2co 6204 Composition of two functio...
oprabco 6205 Composition of a function ...
oprab2co 6206 Composition of operator ab...
df1st2 6207 An alternate possible defi...
df2nd2 6208 An alternate possible defi...
1stconst 6209 The mapping of a restricti...
2ndconst 6210 The mapping of a restricti...
dfmpt2 6211 Alternate definition for t...
curry1 6212 Composition with ` ``' ( 2...
curry1val 6213 The value of a curried fun...
curry1f 6214 Functionality of a curried...
curry2 6215 Composition with ` ``' ( 1...
curry2f 6216 Functionality of a curried...
curry2val 6217 The value of a curried fun...
cnvf1olem 6218 Lemma for ~ cnvf1o . (Con...
cnvf1o 6219 Describe a function that m...
fparlem1 6220 Lemma for ~ fpar . (Contr...
fparlem2 6221 Lemma for ~ fpar . (Contr...
fparlem3 6222 Lemma for ~ fpar . (Contr...
fparlem4 6223 Lemma for ~ fpar . (Contr...
fpar 6224 Merge two functions in par...
fsplit