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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.)
((φ (ψ χ)) → θ)       ((χ (ψ φ)) → θ)

Theoreman32s 502 Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
(((φ ψ) χ) → θ)       (((φ χ) ψ) → θ)

Theoremancom1s 503 Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
(((φ ψ) χ) → θ)       (((ψ φ) χ) → θ)

Theoreman31s 504 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
(((φ ψ) χ) → θ)       (((χ ψ) φ) → θ)

Theoremanass1rs 505 Commutative-associative law for conjunction in an antecedent. (Contributed by Jeff Madsen, 19-Jun-2011.)
((φ (ψ χ)) → θ)       (((φ χ) ψ) → θ)

Theoremanabs1 506 Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(((φ ψ) φ) ↔ (φ ψ))

Theoremanabs5 507 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
((φ (φ ψ)) ↔ (φ ψ))

Theoremanabs7 508 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.)
((ψ (φ ψ)) ↔ (φ ψ))

Theoremanabsan 509 Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) (Revised by NM, 18-Nov-2013.)
(((φ φ) ψ) → χ)       ((φ ψ) → χ)

Theoremanabss1 510 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
(((φ ψ) φ) → χ)       ((φ ψ) → χ)

Theoremanabss4 511 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
(((ψ φ) ψ) → χ)       ((φ ψ) → χ)

Theoremanabss5 512 Absorption of antecedent into conjunction. (Contributed by NM, 10-May-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
((φ (φ ψ)) → χ)       ((φ ψ) → χ)

Theoremanabsi5 513 Absorption of antecedent into conjunction. (Contributed by NM, 11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
(φ → ((φ ψ) → χ))       ((φ ψ) → χ)

Theoremanabsi6 514 Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.)
(φ → ((ψ φ) → χ))       ((φ ψ) → χ)

Theoremanabsi7 515 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
(ψ → ((φ ψ) → χ))       ((φ ψ) → χ)

Theoremanabsi8 516 Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.)
(ψ → ((ψ φ) → χ))       ((φ ψ) → χ)

Theoremanabss7 517 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 19-Nov-2013.)
((ψ (φ ψ)) → χ)       ((φ ψ) → χ)

Theoremanabsan2 518 Absorption of antecedent with conjunction. (Contributed by NM, 10-May-2004.) (Revised by NM, 1-Jan-2013.)
((φ (ψ ψ)) → χ)       ((φ ψ) → χ)

Theoremanabss3 519 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
(((φ ψ) ψ) → χ)       ((φ ψ) → χ)

Theoreman4 520 Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)
(((φ ψ) (χ θ)) ↔ ((φ χ) (ψ θ)))

Theoreman42 521 Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
(((φ ψ) (χ θ)) ↔ ((φ χ) (θ ψ)))

Theoreman4s 522 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
(((φ ψ) (χ θ)) → τ)       (((φ χ) (ψ θ)) → τ)

Theoreman42s 523 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
(((φ ψ) (χ θ)) → τ)       (((φ χ) (θ ψ)) → τ)

Theoremanandi 524 Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
((φ (ψ χ)) ↔ ((φ ψ) (φ χ)))

Theoremanandir 525 Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
(((φ ψ) χ) ↔ ((φ χ) (ψ χ)))

Theoremanandis 526 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
(((φ ψ) (φ χ)) → τ)       ((φ (ψ χ)) → τ)

Theoremanandirs 527 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
(((φ χ) (ψ χ)) → τ)       (((φ ψ) χ) → τ)

Theoremimpbida 528 Deduce an equivalence from two implications. (Contributed by NM, 17-Feb-2007.)
((φ ψ) → χ)    &   ((φ χ) → ψ)       (φ → (ψχ))

Theorempm3.45 529 Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((φψ) → ((φ χ) → (ψ χ)))

Theoremim2anan9 530 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
(φ → (ψχ))    &   (θ → (τη))       ((φ θ) → ((ψ τ) → (χ η)))

Theoremim2anan9r 531 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
(φ → (ψχ))    &   (θ → (τη))       ((θ φ) → ((ψ τ) → (χ η)))

Theoremanim12dan 532 Conjoin antecedents and consequents in a deduction. (Contributed by Mario Carneiro, 12-May-2014.)
((φ ψ) → χ)    &   ((φ θ) → τ)       ((φ (ψ θ)) → (χ τ))

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.)
((φ ψ) → (φψ))

Theorempm3.43 534 Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 27-Nov-2013.)
(((φψ) (φχ)) → (φ → (ψ χ)))

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.)
((φ → (ψ χ)) ↔ ((φψ) (φχ)))

Theorempm4.76 536 Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
(((φψ) (φχ)) ↔ (φ → (ψ χ)))

Theorempm4.38 537 Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
(((φχ) (ψθ)) → ((φ ψ) ↔ (χ θ)))

Theorembi2anan9 538 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 31-Jul-1995.)
(φ → (ψχ))    &   (θ → (τη))       ((φ θ) → ((ψ τ) ↔ (χ η)))

Theorembi2anan9r 539 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.)
(φ → (ψχ))    &   (θ → (τη))       ((θ φ) → ((ψ τ) ↔ (χ η)))

Theorembi2bian9 540 Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)
(φ → (ψχ))    &   (θ → (τη))       ((φ θ) → ((ψτ) ↔ (χη)))

Theorempm5.33 541 Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((φ (ψχ)) ↔ (φ ((φ ψ) → χ)))

Theorempm5.36 542 Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((φ (φψ)) ↔ (ψ (φψ)))

Theorembianabs 543 Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.)
(φ → (ψ ↔ (φ χ)))       (φ → (ψχ))

1.2.5  Logical negation (intuitionistic)

Axiomax-in1 544 'Not' introduction. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
((φ → ¬ φ) → ¬ φ)

Axiomax-in2 545 'Not' elimination. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
φ → (φψ))

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 750 which only holds for some propositions. (Contributed by Mario Carneiro, 12-May-2015.)
((φ → ¬ φ) → ¬ φ)

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.)
φ → (φψ))

Theorempm2.01d 548 Deduction based on reductio ad absurdum. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
(φ → (ψ → ¬ ψ))       (φ → ¬ ψ)

Theorempm2.21d 549 A contradiction implies anything. Deduction from pm2.21 547. (Contributed by NM, 10-Feb-1996.)
(φ → ¬ ψ)       (φ → (ψχ))

Theorempm2.21dd 550 A contradiction implies anything. Deduction from pm2.21 547. (Contributed by Mario Carneiro, 9-Feb-2017.)
(φψ)    &   (φ → ¬ ψ)       (φχ)

Theorempm2.24 551 Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
(φ → (¬ φψ))

Theorempm2.24d 552 Deduction version of pm2.24 551. (Contributed by NM, 30-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
(φψ)       (φ → (¬ ψχ))

Theorempm2.24i 553 Inference version of pm2.24 551. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
φ       φψ)

Theoremcon2d 554 A contraposition deduction. (Contributed by NM, 19-Aug-1993.) (Revised by NM, 12-Feb-2013.)
(φ → (ψ → ¬ χ))       (φ → (χ → ¬ ψ))

Theoremmt2d 555 Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
(φχ)    &   (φ → (ψ → ¬ χ))       (φ → ¬ ψ)

Theoremnsyl3 556 A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) (Revised by NM, 13-Jun-2013.)
(φ → ¬ ψ)    &   (χψ)       (χ → ¬ φ)

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.)
(φ → ¬ ψ)       (ψ → ¬ φ)

Theoremnsyl 558 A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
(φ → ¬ ψ)    &   (χψ)       (φ → ¬ χ)

Theoremnotnot 559 Double negation introduction. Theorem *2.12 of [WhiteheadRussell] p. 101. This one holds for all propositions, but its converse only holds for decidable propositions (see notnotrdc 751). (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
(φ → ¬ ¬ φ)

Theoremnotnotd 560 Deduction associated with notnot 559 and notnoti 574. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
(φψ)       (φ → ¬ ¬ ψ)

Theoremcon3d 561 A contraposition deduction. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)
(φ → (ψχ))       (φ → (¬ χ → ¬ ψ))

Theoremcon3i 562 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
(φψ)       ψ → ¬ φ)

Theoremcon3rr3 563 Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
(φ → (ψχ))       χ → (φ → ¬ ψ))

Theoremcon3and 564 Variant of con3d 561 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(φ → (ψχ))       ((φ ¬ χ) → ¬ ψ)

Theorempm2.01da 565 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
((φ ψ) → ¬ ψ)       (φ → ¬ ψ)

Theorempm3.2im 566 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.)
(φ → (ψ → ¬ (φ → ¬ ψ)))

Theoremexpi 567 An exportation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)
(¬ (φ → ¬ ψ) → χ)       (φ → (ψχ))

Theorempm2.65i 568 Inference rule for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
(φψ)    &   (φ → ¬ ψ)        ¬ φ

Theoremmt2 569 A rule similar to modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.)
ψ    &   (φ → ¬ ψ)        ¬ φ

Theorembiijust 570 Theorem used to justify definition of intuitionistic biconditional df-bi 110. (Contributed by NM, 24-Nov-2017.)
((((φψ) (ψφ)) → ((φψ) (ψφ))) (((φψ) (ψφ)) → ((φψ) (ψφ))))

Theoremcon3 571 Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Feb-2013.)
((φψ) → (¬ ψ → ¬ φ))

Theoremcon2 572 Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2013.)
((φ → ¬ ψ) → (ψ → ¬ φ))

Theoremmt2i 573 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
χ    &   (φ → (ψ → ¬ χ))       (φ → ¬ ψ)

Theoremnotnoti 574 Infer double negation. (Contributed by NM, 27-Feb-2008.)
φ        ¬ ¬ φ

Theorempm2.21i 575 A contradiction implies anything. Inference from pm2.21 547. (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ φ       (φψ)

Theorempm2.24ii 576 A contradiction implies anything. Inference from pm2.24 551. (Contributed by NM, 27-Feb-2008.)
φ    &    ¬ φ       ψ

Theoremnsyld 577 A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
(φ → (ψ → ¬ χ))    &   (φ → (τχ))       (φ → (ψ → ¬ τ))

Theoremnsyli 578 A negated syllogism inference. (Contributed by NM, 3-May-1994.)
(φ → (ψχ))    &   (θ → ¬ χ)       (φ → (θ → ¬ ψ))

Theoremmth8 579 Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
(φ → (¬ ψ → ¬ (φψ)))

Theoremjc 580 Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (φχ)       (φ → ¬ (ψ → ¬ χ))

Theorempm2.51 581 Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (φψ) → (φ → ¬ ψ))

Theorempm2.52 582 Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
(¬ (φψ) → (¬ φ → ¬ ψ))

Theoremexpt 583 Exportation theorem expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.)
((¬ (φ → ¬ ψ) → χ) → (φ → (ψχ)))

Theoremjarl 584 Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.)
(((φψ) → χ) → (¬ φχ))

Theorempm2.65 585 Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. Proofs, such as this one, which assume a proposition, here φ, derive a contradiction, and therefore conclude ¬ φ, 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 ¬ φ, derive a contradiction, and conclude φ, such as condandc 775, are only valid for decidable propositions. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.)
((φψ) → ((φ → ¬ ψ) → ¬ φ))

Theorempm2.65d 586 Deduction rule for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.)
(φ → (ψχ))    &   (φ → (ψ → ¬ χ))       (φ → ¬ ψ)

Theorempm2.65da 587 Deduction rule for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
((φ ψ) → χ)    &   ((φ ψ) → ¬ χ)       (φ → ¬ ψ)

Theoremmto 588 The rule of modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
¬ ψ    &   (φψ)        ¬ φ

Theoremmtod 589 Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
(φ → ¬ χ)    &   (φ → (ψχ))       (φ → ¬ ψ)

Theoremmtoi 590 Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
¬ χ    &   (φ → (ψχ))       (φ → ¬ ψ)

Theoremmtand 591 A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
(φ → ¬ χ)    &   ((φ ψ) → χ)       (φ → ¬ ψ)

Theoremnotbid 592 Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(φ → (ψχ))       (φ → (¬ ψ ↔ ¬ χ))

Theoremcon2b 593 Contraposition. Bidirectional version of con2 572. (Contributed by NM, 5-Aug-1993.)
((φ → ¬ ψ) ↔ (ψ → ¬ φ))

Theoremnotbii 594 Negate both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
(φψ)       φ ↔ ¬ ψ)

Theoremmtbi 595 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
¬ φ    &   (φψ)        ¬ ψ

Theoremmtbir 596 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 14-Oct-2012.)
¬ ψ    &   (φψ)        ¬ φ

Theoremmtbid 597 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
(φ → ¬ ψ)    &   (φ → (ψχ))       (φ → ¬ χ)

Theoremmtbird 598 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
(φ → ¬ χ)    &   (φ → (ψχ))       (φ → ¬ ψ)

Theoremmtbii 599 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
¬ ψ    &   (φ → (ψχ))       (φ → ¬ χ)

Theoremmtbiri 600 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
¬ χ    &   (φ → (ψχ))       (φ → ¬ ψ)

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