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Theorem List for Intuitionistic Logic Explorer - 501-600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoreman13s 501 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
   =>   
 
Theoreman32s 502 Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
   =>   
 
Theoremancom1s 503 Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
   =>   
 
Theoreman31s 504 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
   =>   
 
Theoremanass1rs 505 Commutative-associative law for conjunction in an antecedent. (Contributed by Jeff Madsen, 19-Jun-2011.)
   =>   
 
Theoremanabs1 506 Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
 
Theoremanabs5 507 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
 
Theoremanabs7 508 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.)
 
Theoremanabsan 509 Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) (Revised by NM, 18-Nov-2013.)
   =>   
 
Theoremanabss1 510 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
   =>   
 
Theoremanabss4 511 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
   =>   
 
Theoremanabss5 512 Absorption of antecedent into conjunction. (Contributed by NM, 10-May-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
   =>   
 
Theoremanabsi5 513 Absorption of antecedent into conjunction. (Contributed by NM, 11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
   =>   
 
Theoremanabsi6 514 Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.)
   =>   
 
Theoremanabsi7 515 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
   =>   
 
Theoremanabsi8 516 Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.)
   =>   
 
Theoremanabss7 517 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 19-Nov-2013.)
   =>   
 
Theoremanabsan2 518 Absorption of antecedent with conjunction. (Contributed by NM, 10-May-2004.) (Revised by NM, 1-Jan-2013.)
   =>   
 
Theoremanabss3 519 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
   =>   
 
Theoreman4 520 Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)
 
Theoreman42 521 Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
 
Theoreman4s 522 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
   =>   
 
Theoreman42s 523 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
   =>   
 
Theoremanandi 524 Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
 
Theoremanandir 525 Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
 
Theoremanandis 526 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
   =>   
 
Theoremanandirs 527 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
   =>   
 
Theoremimpbida 528 Deduce an equivalence from two implications. (Contributed by NM, 17-Feb-2007.)
   &       =>   
 
Theorempm3.45 529 Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
 
Theoremim2anan9 530 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
   &       =>   
 
Theoremim2anan9r 531 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
   &       =>   
 
Theoremanim12dan 532 Conjoin antecedents and consequents in a deduction. (Contributed by Mario Carneiro, 12-May-2014.)
   &       =>   
 
Theorempm5.1 533 Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.)
 
Theorempm3.43 534 Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 27-Nov-2013.)
 
Theoremjcab 535 Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 27-Nov-2013.)
 
Theorempm4.76 536 Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
 
Theorempm4.38 537 Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
 
Theorembi2anan9 538 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 31-Jul-1995.)
   &       =>   
 
Theorembi2anan9r 539 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.)
   &       =>   
 
Theorembi2bian9 540 Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)
   &       =>   
 
Theorempm5.33 541 Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 
Theorempm5.36 542 Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 
Theorembianabs 543 Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.)
   =>   
 
1.2.5  Logical negation (intuitionistic)
 
Axiomax-in1 544 'Not' introduction. Axiom 4 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
 
Axiomax-in2 545 'Not' elimination. Axiom 5 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
 
Theorempm2.01 546 Reductio ad absurdum. Theorem *2.01 of [WhiteheadRussell] p. 100. This is valid intuitionistically (in the terminology of Section 1.2 of [Bauer] p. 482 it is a proof of negation not a proof by contradiction); compare with pm2.18dc 749 which only holds for some propositions. (Contributed by Mario Carneiro, 12-May-2015.)
 
Theorempm2.21 547 From a wff and its negation, anything is true. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. (Contributed by Mario Carneiro, 12-May-2015.)
 
Theorempm2.01d 548 Deduction based on reductio ad absurdum. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
   =>   
 
Theorempm2.21d 549 A contradiction implies anything. Deduction from pm2.21 547. (Contributed by NM, 10-Feb-1996.)
   =>   
 
Theorempm2.21dd 550 A contradiction implies anything. Deduction from pm2.21 547. (Contributed by Mario Carneiro, 9-Feb-2017.)
   &       =>   
 
Theorempm2.24 551 Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
 
Theorempm2.24d 552 Deduction version of pm2.24 551. (Contributed by NM, 30-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
   =>   
 
Theorempm2.24i 553 Inference version of pm2.24 551. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
   =>   
 
Theoremcon2d 554 A contraposition deduction. (Contributed by NM, 19-Aug-1993.) (Revised by NM, 12-Feb-2013.)
   =>   
 
Theoremmt2d 555 Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
   &       =>   
 
Theoremnsyl3 556 A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) (Revised by NM, 13-Jun-2013.)
   &       =>   
 
Theoremcon2i 557 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 13-Jun-2013.)
   =>   
 
Theoremnsyl 558 A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
   &       =>   
 
Theoremnotnot1 559 Adding double negation. Theorem *2.12 of [WhiteheadRussell] p. 101. This one holds intuitionistically, but its converse does not (see notnot2dc 750). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
 
Theoremcon3d 560 A contraposition deduction. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)
   =>   
 
Theoremcon3i 561 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
   =>   
 
Theoremcon3rr3 562 Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
   =>   
 
Theoremcon3and 563 Variant of con3d 560 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
   =>   
 
Theorempm2.01da 564 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
   =>   
 
Theorempm3.2im 565 In classical logic, this is just a restatement of pm3.2 126. In intuitionistic logic, it still holds, but is weaker than pm3.2. (Contributed by Mario Carneiro, 12-May-2015.)
 
Theoremexpi 566 An exportation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)
   =>   
 
Theorempm2.65i 567 Inference rule for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
   &       =>   
 
Theoremmt2 568 A rule similar to modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.)
   &       =>   
 
Theorembiijust 569 Theorem used to justify definition of intuitionistic biconditional df-bi 110. (Contributed by NM, 24-Nov-2017.)
 
Theoremcon3 570 Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Feb-2013.)
 
Theoremcon2 571 Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2013.)
 
Theoremmt2i 572 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
   &       =>   
 
Theoremnotnoti 573 Infer double negation. (Contributed by NM, 27-Feb-2008.)
   =>   
 
Theorempm2.21i 574 A contradiction implies anything. Inference from pm2.21 547. (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
   =>   
 
Theorempm2.24ii 575 A contradiction implies anything. Inference from pm2.24 551. (Contributed by NM, 27-Feb-2008.)
   &       =>   
 
Theoremnsyld 576 A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
   &       =>   
 
Theoremnsyli 577 A negated syllogism inference. (Contributed by NM, 3-May-1994.)
   &       =>   
 
Theoremmth8 578 Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
 
Theoremjc 579 Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)
   &       =>   
 
Theorempm2.51 580 Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 
Theorempm2.52 581 Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 
Theoremexpt 582 Exportation theorem expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.)
 
Theoremjarl 583 Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.)
 
Theorempm2.65 584 Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. Proofs, such as this one, which assume a proposition, here , derive a contradiction, and therefore conclude , are valid intuitionistically (and can be called "proof of negation", for example by Section 1.2 of [Bauer] p. 482). By contrast, proofs which assume , derive a contradiction, and conclude , such as condandc 774, are only valid for decidable propositions. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.)
 
Theorempm2.65d 585 Deduction rule for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.)
   &       =>   
 
Theorempm2.65da 586 Deduction rule for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
   &       =>   
 
Theoremmto 587 The rule of modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
   &       =>   
 
Theoremmtod 588 Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
   &       =>   
 
Theoremmtoi 589 Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
   &       =>   
 
Theoremmtand 590 A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
   &       =>   
 
Theoremnotbid 591 Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
   =>   
 
Theoremcon2b 592 Contraposition. Bidirectional version of con2 571. (Contributed by NM, 5-Aug-1993.)
 
Theoremnotbii 593 Negate both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
   =>   
 
Theoremmtbi 594 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
   &       =>   
 
Theoremmtbir 595 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 14-Oct-2012.)
   &       =>   
 
Theoremmtbid 596 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
   &       =>   
 
Theoremmtbird 597 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
   &       =>   
 
Theoremmtbii 598 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
   &       =>   
 
Theoremmtbiri 599 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
   &       =>   
 
Theoremsylnib 600 A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
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