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Theorem List for Intuitionistic Logic Explorer - 301-400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembiantrur 301 A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.)

Theorembiantrud 302 A wff is equivalent to its conjunction with truth. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.)

Theorembiantrurd 303 A wff is equivalent to its conjunction with truth. (Contributed by NM, 1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theoremjca 304 Deduce conjunction of the consequents of two implications ("join consequents with 'and'"). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)

Theoremjcad 305 Deduction conjoining the consequents of two implications. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)

Theoremjca2 306 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Theoremjca31 307 Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)

Theoremjca32 308 Join three consequents. (Contributed by FL, 1-Aug-2009.)

Theoremjcai 309 Deduction replacing implication with conjunction. (Contributed by NM, 5-Aug-1993.)

Theoremjctil 310 Inference conjoining a theorem to left of consequent in an implication. (Contributed by NM, 31-Dec-1993.)

Theoremjctir 311 Inference conjoining a theorem to right of consequent in an implication. (Contributed by NM, 31-Dec-1993.)

Theoremjctl 312 Inference conjoining a theorem to the left of a consequent. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)

Theoremjctr 313 Inference conjoining a theorem to the right of a consequent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)

Theoremjctild 314 Deduction conjoining a theorem to left of consequent in an implication. (Contributed by NM, 21-Apr-2005.)

Theoremjctird 315 Deduction conjoining a theorem to right of consequent in an implication. (Contributed by NM, 21-Apr-2005.)

Theoremancl 316 Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)

Theoremanclb 317 Conjoin antecedent to left of consequent. Theorem *4.7 of [WhiteheadRussell] p. 120. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)

Theorempm5.42 318 Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)

Theoremancr 319 Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)

Theoremancrb 320 Conjoin antecedent to right of consequent. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)

Theoremancli 321 Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.)

Theoremancri 322 Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)

Theoremancld 323 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)

Theoremancrd 324 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)

Theoremanc2l 325 Conjoin antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 14-Jul-2013.)

Theoremanc2r 326 Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.)

Theoremanc2li 327 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)

Theoremanc2ri 328 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)

Theorempm3.41 329 Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)

Theorempm3.42 330 Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)

Theorempm3.4 331 Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)

Theorempm4.45im 332 Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)

Theoremanim12d 333 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 18-Dec-2013.)

Theoremanim1d 334 Add a conjunct to right of antecedent and consequent in a deduction. (Contributed by NM, 3-Apr-1994.)

Theoremanim2d 335 Add a conjunct to left of antecedent and consequent in a deduction. (Contributed by NM, 5-Aug-1993.)

Theoremanim12i 336 Conjoin antecedents and consequents of two premises. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Dec-2013.)

Theoremanim12ci 337 Variant of anim12i 336 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremanim1i 338 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)

Theoremanim1ci 339 Introduce conjunct to both sides of an implication. (Contributed by Peter Mazsa, 24-Sep-2022.)

Theoremanim2i 340 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)

Theoremanim12ii 341 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.)

Theoremanim12 342 Theorem *3.47 of [WhiteheadRussell] p. 113. It was proved by Leibniz, and it evidently pleased him enough to call it 'praeclarum theorema' (splendid theorem). (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Theorempm3.33 343 Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)

Theorempm3.34 344 Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)

Theorempm3.35 345 Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. (Contributed by NM, 14-Dec-2002.)

Theorempm5.31 346 Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)

Theoremimp4a 347 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp4b 348 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp4c 349 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp4d 350 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp41 351 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp42 352 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp43 353 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp44 354 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp45 355 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp5a 356 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremimp5d 357 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremimp5g 358 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremimp55 359 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremimp511 360 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexpimpd 361 Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.)

Theoremexp31 362 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp32 363 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp4a 364 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp4b 365 An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)

Theoremexp4c 366 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp4d 367 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp41 368 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp42 369 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp43 370 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp44 371 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp45 372 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexpr 373 Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremexp5c 374 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp53 375 An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremexpl 376 Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.)

Theoremimpr 377 Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremimpl 378 Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremimpac 379 Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.)

Theoremexbiri 380 Inference form of exbir 1416. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.)

Theoremsimprbda 381 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)

Theoremsimplbda 382 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)

Theoremsimplbi2 383 Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.)

Theoremsimpl2im 384 Implication from an eliminated conjunct implied by the antecedent. (Contributed by BJ/AV, 5-Apr-2021.)

Theoremsimplbiim 385 Implication from an eliminated conjunct equivalent to the antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremdfbi2 386 A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)

Theorempm4.71 387 Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 2-Dec-2012.)

Theorempm4.71r 388 Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 25-Jul-1999.)

Theorempm4.71i 389 Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 4-Jan-2004.)

Theorempm4.71ri 390 Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 1-Dec-2003.)

Theorempm4.71d 391 Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by Mario Carneiro, 25-Dec-2016.)

Theorempm4.71rd 392 Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 10-Feb-2005.)

Theorempm4.24 393 Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 14-Mar-2014.)

Theoremanidm 394 Idempotent law for conjunction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)

Theoremanidms 395 Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.)

Theoremanidmdbi 396 Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)

Theoremanasss 397 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)

Theoremanassrs 398 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)

Theoremanass 399 Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)

Theoremsylanl1 400 A syllogism inference. (Contributed by NM, 10-Mar-2005.)

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