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Theorem List for Intuitionistic Logic Explorer - 401-500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmpan2 401 An inference based on modus ponens. (The proof was shortened by Wolf Lammen, 19-Nov-2012.)
   &       =>   
 
Theoremmp2an 402 An inference based on modus ponens.
   &       &       =>   
 
Theoremmp4an 403 An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2011.)
   &       &       &       &       =>   
 
Theoremmpan2d 404 A deduction based on modus ponens.
   &       =>   
 
Theoremmpand 405 A deduction based on modus ponens. (The proof was shortened by Wolf Lammen, 7-Apr-2013.)
   &       =>   
 
Theoremmpani 406 An inference based on modus ponens. (The proof was shortened by Wolf Lammen, 19-Nov-2012.)
   &       =>   
 
Theoremmpan2i 407 An inference based on modus ponens. (The proof was shortened by Wolf Lammen, 19-Nov-2012.)
   &       =>   
 
Theoremmp2ani 408 An inference based on modus ponens.
   &       &       =>   
 
Theoremmp2and 409 A deduction based on modus ponens.
   &       &       =>   
 
Theoremmpanl1 410 An inference based on modus ponens. (The proof was shortened by Wolf Lammen, 7-Apr-2013.)
   &       =>   
 
Theoremmpanl2 411 An inference based on modus ponens. (The proof was shortened by Andrew Salmon, 7-May-2011.)
   &       =>   
 
Theoremmpanl12 412 An inference based on modus ponens.
   &       &       =>   
 
Theoremmpanr1 413 An inference based on modus ponens. (The proof was shortened by Andrew Salmon, 7-May-2011.)
   &       =>   
 
Theoremmpanr2 414 An inference based on modus ponens. (The proof was shortened by Andrew Salmon, 7-May-2011.) (The proof was shortened by Wolf Lammen, 7-Apr-2013.)
   &       =>   
 
Theoremmpanr2OLD 415 Obsolete proof of mpanr2 414 as of 7-Apr-2013.)
   &       =>   
 
Theoremmpanr12 416 An inference based on modus ponens.
   &       &       =>   
 
Theoremmpanlr1 417 An inference based on modus ponens. (The proof was shortened by Wolf Lammen, 7-Apr-2013.)
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Theorempm5.74da 418 Distribution of implication over biconditional (deduction rule).
   =>   
 
Theoremimdistan 419 Distribution of implication with conjunction. (The proof was shortened by Wolf Lammen, 6-Dec-2012.)
 
Theoremimdistani 420 Distribution of implication with conjunction.
   =>   
 
Theoremimdistanri 421 Distribution of implication with conjunction.
   =>   
 
Theoremimdistand 422 Distribution of implication with conjunction (deduction rule).
   =>   
 
Theoremimdistanda 423 Distribution of implication with conjunction (deduction version with conjoined antecedent). (Contributed by Jeff Madsen, 19-Jun-2011.)
   =>   
 
Theorempm5.32d 424 Distribution of implication over biconditional (deduction rule).
   =>   
 
Theorempm5.32rd 425 Distribution of implication over biconditional (deduction rule).
   =>   
 
Theorempm5.32da 426 Distribution of implication over biconditional (deduction rule).
   =>   
 
Theorempm5.32 427 Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125.
 
Theorempm5.32i 428 Distribution of implication over biconditional (inference rule).
   =>   
 
Theorempm5.32ri 429 Distribution of implication over biconditional (inference rule).
   =>   
 
Theorembiadan2 430 Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.)
   &       =>   
 
Theoremanbi2i 431 Introduce a left conjunct to both sides of a logical equivalence. (The proof was shortened by Wolf Lammen, 16-Nov-2013.)
   =>   
 
Theoremanbi1i 432 Introduce a right conjunct to both sides of a logical equivalence. (The proof was shortened by Wolf Lammen, 16-Nov-2013.)
   =>   
 
Theoremanbi12i 433 Conjoin both sides of two equivalences.
   &       =>   
 
Theoremsylan9bb 434 Nested syllogism inference conjoining dissimilar antecedents.
   &       =>   
 
Theoremsylan9bbr 435 Nested syllogism inference conjoining dissimilar antecedents.
   &       =>   
 
Theoremanbi2d 436 Deduction adding a left conjunct to both sides of a logical equivalence. (The proof was shortened by Wolf Lammen, 16-Nov-2013.)
   =>   
 
Theoremanbi1d 437 Deduction adding a right conjunct to both sides of a logical equivalence. (The proof was shortened by Wolf Lammen, 16-Nov-2013.)
   =>   
 
Theoremanbi1 438 Introduce a right conjunct to both sides of a logical equivalence. Theorem *4.36 of [WhiteheadRussell] p. 118.
 
Theoremanbi2 439 Introduce a left conjunct to both sides of a logical equivalence.
 
Theorembitr 440 Theorem *4.22 of [WhiteheadRussell] p. 117.
 
Theoremanbi12d 441 Deduction joining two equivalences to form equivalence of conjunctions.
   &       =>   
 
Theorempm5.3 442 Theorem *5.3 of [WhiteheadRussell] p. 125. (The proof was shortened by Andrew Salmon, 7-May-2011.)
 
Theoremadantll 443 Deduction adding a conjunct to antecedent. (The proof was shortened by Wolf Lammen, 24-Nov-2012.)
   =>   
 
Theoremadantlr 444 Deduction adding a conjunct to antecedent. (The proof was shortened by Wolf Lammen, 24-Nov-2012.)
   =>   
 
Theoremadantrl 445 Deduction adding a conjunct to antecedent. (The proof was shortened by Wolf Lammen, 24-Nov-2012.)
   =>   
 
Theoremadantrr 446 Deduction adding a conjunct to antecedent. (The proof was shortened by Wolf Lammen, 24-Nov-2012.)
   =>   
 
Theoremadantlll 447 Deduction adding a conjunct to antecedent. (The proof was shortened by Wolf Lammen, 2-Dec-2012.)
   =>   
 
Theoremadantllr 448 Deduction adding a conjunct to antecedent. (The proof was shortened by Wolf Lammen, 4-Dec-2012.)
   =>   
 
Theoremadantlrl 449 Deduction adding a conjunct to antecedent. (The proof was shortened by Wolf Lammen, 4-Dec-2012.)
   =>   
 
Theoremadantlrr 450 Deduction adding a conjunct to antecedent. (The proof was shortened by Wolf Lammen, 4-Dec-2012.)
   =>   
 
Theoremadantrll 451 Deduction adding a conjunct to antecedent. (The proof was shortened by Wolf Lammen, 4-Dec-2012.)
   =>   
 
Theoremadantrlr 452 Deduction adding a conjunct to antecedent. (The proof was shortened by Wolf Lammen, 4-Dec-2012.)
   =>   
 
Theoremadantrrl 453 Deduction adding a conjunct to antecedent. (The proof was shortened by Wolf Lammen, 4-Dec-2012.)
   =>   
 
Theoremadantrrr 454 Deduction adding a conjunct to antecedent. (The proof was shortened by Wolf Lammen, 4-Dec-2012.)
   =>   
 
Theoremad2antrr 455 Deduction adding conjuncts to antecedent. (The proof was shortened by Wolf Lammen, 20-Nov-2012.)
   =>   
 
Theoremad3antrrr 456 Deduction adding 3 conjuncts to antecedent.
   =>   
 
Theoremad2antlr 457 Deduction adding conjuncts to antecedent. (The proof was shortened by Wolf Lammen, 20-Nov-2012.)
   =>   
 
Theoremad2antrl 458 Deduction adding conjuncts to antecedent.
   =>   
 
Theoremad2antll 459 Deduction adding conjuncts to antecedent.
   =>   
 
Theoremad2ant2l 460 Deduction adding two conjuncts to antecedent.
   =>   
 
Theoremad2ant2r 461 Deduction adding two conjuncts to antecedent.
   =>   
 
Theoremad2ant2lr 462 Deduction adding two conjuncts to antecedent.
   =>   
 
Theoremad2ant2rl 463 Deduction adding two conjuncts to antecedent.
   =>   
 
Theoremsimpll 464 Simplification of a conjunction.
 
Theoremsimplr 465 Simplification of a conjunction.
 
Theoremsimprl 466 Simplification of a conjunction.
 
Theoremsimprr 467 Simplification of a conjunction.
 
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. (The proof was 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". (The proof was 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.
 
Theoreman32 479 A rearrangement of conjuncts. (The proof was shortened by Wolf Lammen, 25-Dec-2012.)
 
Theoreman13 480 A rearrangement of conjuncts. (The proof was shortened by Wolf Lammen, 31-Dec-2012.)
 
Theoreman31 481 A rearrangement of conjuncts. (The proof was 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 14 (or a variant).
   =>   
 
Theoremancom2s 483 Inference commuting a nested conjunction in antecedent. (The proof was shortened by Wolf Lammen, 24-Nov-2012.)
   =>   
 
Theoreman13s 484 Swap two conjuncts in antecedent.
   =>   
 
Theoreman32s 485 Swap two conjuncts in antecedent.
   =>   
 
Theoremancom1s 486 Inference commuting a nested conjunction in antecedent. (The proof was shortened by Wolf Lammen, 24-Nov-2012.)
   =>   
 
Theoreman31s 487 Swap two conjuncts in antecedent.
   =>   
 
Theoremanabs1 488 Absorption into embedded conjunct. (The proof was shortened by Wolf Lammen, 16-Nov-2013.)
 
Theoremanabs5 489 Absorption into embedded conjunct. (The proof was shortened by Wolf Lammen, 9-Dec-2012.)
 
Theoremanabs7 490 Absorption into embedded conjunct. (The proof was shortened by Wolf Lammen, 17-Nov-2013.)
 
Theoremanabsan 491 Absorption of antecedent with conjunction.
   =>   
 
Theoremanabss1 492 Absorption of antecedent into conjunction. (The proof was shortened by Wolf Lammen, 31-Dec-2012.)
   =>   
 
Theoremanabss4 493 Absorption of antecedent into conjunction.
   =>   
 
Theoremanabss5 494 Absorption of antecedent into conjunction. (The proof was shortened by Wolf Lammen, 1-Jan-2013.)
   =>   
 
Theoremanabsi5 495 Absorption of antecedent into conjunction. (The proof was shortened by Wolf Lammen, 18-Nov-2013.)
   =>   
 
Theoremanabsi6 496 Absorption of antecedent into conjunction.
   =>   
 
Theoremanabsi7 497 Absorption of antecedent into conjunction. (The proof was shortened by Wolf Lammen, 18-Nov-2013.)
   =>   
 
Theoremanabsi8 498 Absorption of antecedent into conjunction.
   =>   
 
Theoremanabss7 499 Absorption of antecedent into conjunction. (The proof was shortened by Wolf Lammen, 19-Nov-2013.)
   =>   
 
Theoremanabsan2 500 Absorption of antecedent with conjunction.
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