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Theorem List for Intuitionistic Logic Explorer - 201-300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3bitr4ri 201 A chained inference from transitive law for logical equivalence.
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Theorem3bitrd 202 Deduction from transitivity of biconditional.
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Theorem3bitrrd 203 Deduction from transitivity of biconditional.
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Theorem3bitr2d 204 Deduction from transitivity of biconditional.
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Theorem3bitr2rd 205 Deduction from transitivity of biconditional.
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Theorem3bitr3d 206 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula.
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Theorem3bitr3rd 207 Deduction from transitivity of biconditional.
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Theorem3bitr4d 208 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula.
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Theorem3bitr4rd 209 Deduction from transitivity of biconditional.
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Theorem3bitr3g 210 More general version of 3bitr3i 198. Useful for converting definitions in a formula.
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Theorem3bitr4g 211 More general version of 3bitr4i 200. Useful for converting definitions in a formula.
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Theorembi3ant 212 Construct a bi-conditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)
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Theorembisym 213 Express symmetries of theorems in terms of biconditionals. (Contributed by Wolf Lammen, 14-May-2013.)
 
Theoremimbi2i 214 Introduce an antecedent to both sides of a logical equivalence. (The proof was shortened by Wolf Lammen, 6-Feb-2013.)
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Theorembibi2i 215 Inference adding a biconditional to the left in an equivalence. (The proof was shortened by Andrew Salmon, 7-May-2011.) (The proof was shortened by Wolf Lammen, 16-May-2013.)
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Theorembibi1i 216 Inference adding a biconditional to the right in an equivalence.
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Theorembibi12i 217 The equivalence of two equivalences.
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Theoremimbi2d 218 Deduction adding an antecedent to both sides of a logical equivalence.
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Theoremimbi1d 219 Deduction adding a consequent to both sides of a logical equivalence. (The proof was shortened by Wolf Lammen, 17-Sep-2013.)
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Theorembibi2d 220 Deduction adding a biconditional to the left in an equivalence. (The proof was shortened by Wolf Lammen, 19-May-2013.)
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Theorembibi1d 221 Deduction adding a biconditional to the right in an equivalence.
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Theoremimbi12d 222 Deduction joining two equivalences to form equivalence of implications.
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Theorembibi12d 223 Deduction joining two equivalences to form equivalence of biconditionals.
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Theoremimbi1 224 Theorem *4.84 of [WhiteheadRussell] p. 122.
 
Theoremimbi2 225 Theorem *4.85 of [WhiteheadRussell] p. 122. (The proof was shortened by Wolf Lammen, 19-May-2013.)
 
Theoremimbi1i 226 Introduce a consequent to both sides of a logical equivalence. (The proof was shortened by Wolf Lammen, 17-Sep-2013.)
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Theoremimbi12i 227 Join two logical equivalences to form equivalence of implications.
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Theorembibi1 228 Theorem *4.86 of [WhiteheadRussell] p. 122.
 
Theorembiimt 229 A wff is equivalent to itself with true antecedent.
 
Theorempm5.5 230 Theorem *5.5 of [WhiteheadRussell] p. 125.
 
Theorema1bi 231 Inference rule introducing a theorem as an antecedent. The proof was shortened by Wolf Lammen, 11-Nov-2012.)
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Theorempm5.501 232 Theorem *5.501 of [WhiteheadRussell] p. 125.
 
Theoremibib 233 Implication in terms of implication and biconditional. (The proof was shortened by Wolf Lammen, 24-Jan-2013.)
 
Theoremibibr 234 Implication in terms of implication and biconditional. (The proof was shortened by Wolf Lammen, 21-Dec-2013.)
 
Theoremtbt 235 A wff is equivalent to its equivalence with truth. (The proof was shortened by Andrew Salmon, 13-May-2011.)
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Theorembi2.04 236 Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122.
 
Theorempm5.4 237 Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125.
 
Theoremimdi 238 Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125.
 
Theorempm5.41 239 Theorem *5.41 of [WhiteheadRussell] p. 125. (The proof was shortened by Wolf Lammen, 12-Oct-2012.)
 
Theoremimim21b 240 Simplify an implication between two implications when the antecedent of the first is a consequence of the antecedent of the second. The reverse form is useful in producing the successor step in induction proofs. (Contributed by Paul Chapman, 22-Jun-2011.) (The proof was shortened by Wolf Lammen, 14-Sep-2013.)
 
Theoremimp3a 241 Importation deduction.
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Theoremimp31 242 An importation inference.
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Theoremimp32 243 An importation inference.
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Theoremexp3a 244 Exportation deduction.
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Theoremexpdimp 245 A deduction version of exportation, followed by importation.
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Theoremimpancom 246 Mixed importation/commutation inference.
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Theorempm3.3 247 Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112. (The proof was shortened by Wolf Lammen, 24-Mar-2013.)
 
Theorempm3.31 248 Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112. (The proof was shortened by Wolf Lammen, 24-Mar-2013.)
 
Theoremimpexp 249 Import-export theorem. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (The proof was shortened by Wolf Lammen, 24-Mar-2013.)
 
Theorempm3.21 250 Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111.
 
Theorempm3.22 251 Theorem *3.22 of [WhiteheadRussell] p. 111. (The proof was shortened by Wolf Lammen, 13-Nov-2012.)
 
Theoremancom 252 Commutative law for conjunction. Theorem *4.3 of [WhiteheadRussell] p. 118. (The proof was shortened by Wolf Lammen, 4-Nov-2012.)
 
Theoremancomd 253 Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.)
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Theoremancoms 254 Inference commuting conjunction in antecedent.
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Theoremancomsd 255 Deduction commuting conjunction in antecedent.
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Theorempm3.2i 256 Infer conjunction of premises.
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Theorempm3.43i 257 Nested conjunction of antecedents.
 
Theoremsimplbi 258 Deduction eliminating a conjunct.
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Theoremsimprbi 259 Deduction eliminating a conjunct.
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Theoremadantr 260 Inference adding a conjunct to the right of an antecedent.
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Theoremadantl 261 Inference adding a conjunct to the left of an antecedent. (The proof was shortened by Wolf Lammen, 23-Nov-2012.)
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Theoremadantld 262 Deduction adding a conjunct to the left of an antecedent. (The proof was shortened by Wolf Lammen, 20-Dec-2012.)
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Theoremadantrd 263 Deduction adding a conjunct to the right of an antecedent.
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Theoremmpan9 264 Modus ponens conjoining dissimilar antecedents. (The proof was shortened by Andrew Salmon, 7-May-2011.)
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Theoremsyldan 265 A syllogism deduction with conjoined antecents. (The proof was shortened by Wolf Lammen, 6-Apr-2013.)
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Theoremsylan 266 A syllogism inference. (The proof was shortened by Wolf Lammen, 22-Nov-2012.)
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Theoremsylanb 267 A syllogism inference.
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Theoremsylanbr 268 A syllogism inference.
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Theoremsylan2 269 A syllogism inference. (The proof was shortened by Wolf Lammen, 22-Nov-2012.)
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Theoremsylan2b 270 A syllogism inference.
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Theoremsylan2br 271 A syllogism inference.
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Theoremsyl2an 272 A double syllogism inference.
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Theoremsyl2anr 273 A double syllogism inference.
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Theoremsyl2anb 274 A double syllogism inference.
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Theoremsyl2anbr 275 A double syllogism inference.
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Theoremsyland 276 A syllogism deduction.
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Theoremsylan2d 277 A syllogism deduction.
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Theoremsyl2and 278 A syllogism deduction.
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Theorembiimpa 279 Inference from a logical equivalence.
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Theorembiimpar 280 Inference from a logical equivalence.
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Theorembiimpac 281 Inference from a logical equivalence.
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Theorembiimparc 282 Inference from a logical equivalence.
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Theoremiba 283 Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121.
 
Theoremibar 284 Introduction of antecedent as conjunct.
 
Theorembiantru 285 A wff is equivalent to its conjunction with truth.
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Theorembiantrur 286 A wff is equivalent to its conjunction with truth.
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Theorembiantrud 287 A wff is equivalent to its conjunction with truth. (The proof was shortened by Wolf Lammen, 23-Oct-2013.)
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Theorembiantrurd 288 A wff is equivalent to its conjunction with truth. (The proof was shortened by Andrew Salmon, 7-May-2011.)
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Theoremjca 289 Deduce conjunction of the consequents of two implications ("join consequents with 'and'"). (The proof was shortened by Wolf Lammen, 25-Oct-2012.)
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Theoremjcad 290 Deduction conjoining the consequents of two implications. (The proof was shortened by Wolf Lammen, 23-Jul-2013.)
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Theoremjca31 291 Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)
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Theoremjca32 292 Join three consequents. (Contributed by FL, 1-Aug-2009.)
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Theoremjcai 293 Deduction replacing implication with conjunction.
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Theoremjctil 294 Inference conjoining a theorem to left of consequent in an implication.
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Theoremjctir 295 Inference conjoining a theorem to right of consequent in an implication.
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Theoremjctl 296 Inference conjoining a theorem to the left of a consequent. (The proof was shortened by Wolf Lammen, 24-Oct-2012.)
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Theoremjctr 297 Inference conjoining a theorem to the right of a consequent. (The proof was shortened by Wolf Lammen, 24-Oct-2012.)
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Theoremjctild 298 Deduction conjoining a theorem to left of consequent in an implication.
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Theoremjctird 299 Deduction conjoining a theorem to right of consequent in an implication.
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Theoremancl 300 Conjoin antecedent to left of consequent.
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