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Theorem List for Intuitionistic Logic Explorer - 401-500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmp4an 401 An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2011.)
   &       &       &       &       =>   
 
Theoremmpan2d 402 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
   &       =>   
 
Theoremmpand 403 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
   &       =>   
 
Theoremmpani 404 An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
   &       =>   
 
Theoremmpan2i 405 An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
   &       =>   
 
Theoremmp2ani 406 An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)
   &       &       =>   
 
Theoremmp2and 407 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
   &       &       =>   
 
Theoremmpanl1 408 An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
   &       =>   
 
Theoremmpanl2 409 An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)
   &       =>   
 
Theoremmpanl12 410 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
   &       &       =>   
 
Theoremmpanr1 411 An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)
   &       =>   
 
Theoremmpanr2 412 An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
   &       =>   
 
Theoremmpanr2OLD 413 Obsolete proof of mpanr2 412 as of 7-Apr-2013. (Contributed by NM, 3-May-1994.) (Revised by NM, 12-May-2011.)
   &       =>   
 
Theoremmpanr12 414 An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.)
   &       &       =>   
 
Theoremmpanlr1 415 An inference based on modus ponens. (Contributed by NM, 30-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
   &       =>   
 
Theorempm5.74da 416 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 4-May-2007.)
   =>   
 
Theoremimdistan 417 Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.)
 
Theoremimdistani 418 Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)
   =>   
 
Theoremimdistanri 419 Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)
   =>   
 
Theoremimdistand 420 Distribution of implication with conjunction (deduction rule). (Contributed by NM, 27-Aug-2004.)
   =>   
 
Theoremimdistanda 421 Distribution of implication with conjunction (deduction version with conjoined antecedent). (Contributed by Jeff Madsen, 19-Jun-2011.)
   =>   
 
Theorempm5.32d 422 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 29-Oct-1996.) (Revised by NM, 31-Jan-2015.)
   =>   
 
Theorempm5.32rd 423 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 25-Dec-2004.)
   =>   
 
Theorempm5.32da 424 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 9-Dec-2006.)
   =>   
 
Theorempm5.32 425 Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Revised by NM, 31-Jan-2015.)
 
Theorempm5.32i 426 Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)
   =>   
 
Theorempm5.32ri 427 Distribution of implication over biconditional (inference rule). (Contributed by NM, 12-Mar-1995.)
   =>   
 
Theorembiadan2 428 Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.)
   &       =>   
 
Theoremanbi2i 429 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
   =>   
 
Theoremanbi1i 430 Introduce a right conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
   =>   
 
Theoremanbi2ci 431 Variant of anbi2i 429 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
   =>   
 
Theoremanbi12i 432 Conjoin both sides of two equivalences. (Contributed by NM, 5-Aug-1993.)
   &       =>   
 
Theoremsylan9bb 433 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
   &       =>   
 
Theoremsylan9bbr 434 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
   &       =>   
 
Theoremanbi2d 435 Deduction adding a left conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
   =>   
 
Theoremanbi1d 436 Deduction adding a right conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
   =>   
 
Theoremanbi1 437 Introduce a right conjunct to both sides of a logical equivalence. Theorem *4.36 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
 
Theoremanbi2 438 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 16-Nov-2013.)
 
Theorembitr 439 Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
 
Theoremanbi12d 440 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 5-Aug-1993.)
   &       =>   
 
Theoremmpan10 441 Modus ponens mixed with several conjunctions. (Contributed by Jim Kingdon, 7-Jan-2018.)
 
Theorempm5.3 442 Theorem *5.3 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 
Theoremadantll 443 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
   =>   
 
Theoremadantlr 444 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
   =>   
 
Theoremadantrl 445 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
   =>   
 
Theoremadantrr 446 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
   =>   
 
Theoremadantlll 447 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 2-Dec-2012.)
   =>   
 
Theoremadantllr 448 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
   =>   
 
Theoremadantlrl 449 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
   =>   
 
Theoremadantlrr 450 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
   =>   
 
Theoremadantrll 451 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
   =>   
 
Theoremadantrlr 452 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
   =>   
 
Theoremadantrrl 453 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
   =>   
 
Theoremadantrrr 454 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
   =>   
 
Theoremad2antrr 455 Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.)
   =>   
 
Theoremad3antrrr 456 Deduction adding 3 conjuncts to antecedent. (Contributed by NM, 28-Jul-2012.)
   =>   
 
Theoremad2antlr 457 Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.)
   =>   
 
Theoremad2antrl 458 Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
   =>   
 
Theoremad2antll 459 Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
   =>   
 
Theoremad2ant2l 460 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
   =>   
 
Theoremad2ant2r 461 Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
   =>   
 
Theoremad2ant2lr 462 Deduction adding two conjuncts to antecedent. (Contributed by NM, 23-Nov-2007.)
   =>   
 
Theoremad2ant2rl 463 Deduction adding two conjuncts to antecedent. (Contributed by NM, 24-Nov-2007.)
   =>   
 
Theoremsimpll 464 Simplification of a conjunction. (Contributed by NM, 18-Mar-2007.)
 
Theoremsimplr 465 Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.)
 
Theoremsimprl 466 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
 
Theoremsimprr 467 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
 
Theoremsimplll 468 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 
Theoremsimpllr 469 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 
Theoremsimplrl 470 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 
Theoremsimplrr 471 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 
Theoremsimprll 472 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 
Theoremsimprlr 473 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 
Theoremsimprrl 474 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 
Theoremsimprrr 475 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 
Theorempm4.87 476 Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.)
 
Theoremabai 477 Introduce one conjunct as an antecedent to the other. "abai" stands for "and, biconditional, and, implication". (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
 
Theoreman12 478 Swap two conjuncts. Note that the first digit (1) in the label refers to the outer conjunct position, and the next digit (2) to the inner conjunct position. (Contributed by NM, 12-Mar-1995.)
 
Theoreman32 479 A rearrangement of conjuncts. (Contributed by NM, 12-Mar-1995.) (Proof shortened by Wolf Lammen, 25-Dec-2012.)
 
Theoreman13 480 A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
 
Theoreman31 481 A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
 
Theoreman12s 482 Swap two conjuncts in antecedent. The label suffix "s" means that an12 478 is combined with syl 13 (or a variant). (Contributed by NM, 13-Mar-1996.)
   =>   
 
Theoremancom2s 483 Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
   =>   
 
Theoreman13s 484 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
   =>   
 
Theoreman32s 485 Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
   =>   
 
Theoremancom1s 486 Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
   =>   
 
Theoreman31s 487 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
   =>   
 
Theoremanabs1 488 Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
 
Theoremanabs5 489 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
 
Theoremanabs7 490 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.)
 
Theoremanabsan 491 Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) (Revised by NM, 18-Nov-2013.)
   =>   
 
Theoremanabss1 492 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
   =>   
 
Theoremanabss4 493 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
   =>   
 
Theoremanabss5 494 Absorption of antecedent into conjunction. (Contributed by NM, 10-May-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
   =>   
 
Theoremanabsi5 495 Absorption of antecedent into conjunction. (Contributed by NM, 11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
   =>   
 
Theoremanabsi6 496 Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.)
   =>   
 
Theoremanabsi7 497 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
   =>   
 
Theoremanabsi8 498 Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.)
   =>   
 
Theoremanabss7 499 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 19-Nov-2013.)
   =>   
 
Theoremanabsan2 500 Absorption of antecedent with conjunction. (Contributed by NM, 10-May-2004.) (Revised by NM, 1-Jan-2013.)
   =>   
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