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

Theoremjctl 301 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 302 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 303 Deduction conjoining a theorem to left of consequent in an implication. (Contributed by NM, 21-Apr-2005.)

Theoremjctird 304 Deduction conjoining a theorem to right of consequent in an implication. (Contributed by NM, 21-Apr-2005.)

Theoremancl 305 Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)

Theoremanclb 306 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.)

Theorempm5.42 307 Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)

Theoremancr 308 Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)

Theoremancrb 309 Conjoin antecedent to right of consequent. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)

Theoremancli 310 Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.)

Theoremancri 311 Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)

Theoremancld 312 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 313 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 314 Conjoin antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 14-Jul-2013.)

Theoremanc2r 315 Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.)

Theoremanc2li 316 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 317 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)

Theorempm3.41 318 Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)

Theorempm3.42 319 Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)

Theorempm3.4 320 Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)

Theorempm4.45im 321 Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)

Theoremanim12d 322 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 18-Dec-2013.)

Theoremanim1d 323 Add a conjunct to right of antecedent and consequent in a deduction. (Contributed by NM, 3-Apr-1994.)

Theoremanim2d 324 Add a conjunct to left of antecedent and consequent in a deduction. (Contributed by NM, 5-Aug-1993.)

Theoremanim12i 325 Conjoin antecedents and consequents of two premises. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Dec-2013.)

Theoremanim12ci 326 Variant of anim12i 325 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremanim1i 327 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)

Theoremanim2i 328 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)

Theoremanim12ii 329 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.)

Theoremprth 330 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.)

Theorempm3.33 331 Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)

Theorempm3.34 332 Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)

Theorempm3.35 333 Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. (Contributed by NM, 14-Dec-2002.)

Theorempm5.31 334 Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)

Theoremimp4a 335 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp4b 336 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp4c 337 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp4d 338 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp41 339 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp42 340 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp43 341 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp44 342 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp45 343 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp5a 344 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremimp5d 345 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremimp5g 346 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremimp55 347 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremimp511 348 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexpimpd 349 Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.)

Theoremexp31 350 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp32 351 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp4a 352 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp4b 353 An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)

Theoremexp4c 354 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp4d 355 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp41 356 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp42 357 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp43 358 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp44 359 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexp45 360 An exportation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexpr 361 Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremexp5c 362 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp53 363 An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremexpl 364 Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.)

Theoremimpr 365 Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremimpl 366 Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremimpac 367 Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.)

Theoremexbiri 368 Inference form of exbir 1341. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.)

Theoremsimprbda 369 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)

Theoremsimplbda 370 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)

Theoremsimplbi2 371 Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.)

Theoremsimpl2im 372 Implication from an eliminated conjunct implied by the antecedent. (Contributed by BJ/AV, 5-Apr-2021.)

Theoremsimplbiim 373 Implication from an eliminated conjunct equivalent to the antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremdfbi2 374 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.)

Theorempm4.71 375 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 376 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 377 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 378 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 379 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 380 Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 10-Feb-2005.)

Theorempm4.24 381 Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 14-Mar-2014.)

Theoremanidm 382 Idempotent law for conjunction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)

Theoremanidms 383 Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.)

Theoremanidmdbi 384 Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)

Theoremanasss 385 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)

Theoremanassrs 386 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)

Theoremanass 387 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 388 A syllogism inference. (Contributed by NM, 10-Mar-2005.)

Theoremsylanl2 389 A syllogism inference. (Contributed by NM, 1-Jan-2005.)

Theoremsylanr1 390 A syllogism inference. (Contributed by NM, 9-Apr-2005.)

Theoremsylanr2 391 A syllogism inference. (Contributed by NM, 9-Apr-2005.)

Theoremsylani 392 A syllogism inference. (Contributed by NM, 2-May-1996.)

Theoremsylan2i 393 A syllogism inference. (Contributed by NM, 1-Aug-1994.)

Theoremsyl2ani 394 A syllogism inference. (Contributed by NM, 3-Aug-1999.)

Theoremsylan9 395 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theoremsylan9r 396 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.)

Theoremsyl2anc 397 Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.)

Theoremsylancl 398 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsylancr 399 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsylanblc 400 Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)

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