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Theorem List for Intuitionistic Logic Explorer - 501-600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoreman4 501 Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)
(((φ ψ) (χ θ)) ↔ ((φ χ) (ψ θ)))

Theoreman42 502 Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
(((φ ψ) (χ θ)) ↔ ((φ χ) (θ ψ)))

Theoreman4s 503 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
(((φ ψ) (χ θ)) → τ)       (((φ χ) (ψ θ)) → τ)

Theoreman42s 504 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
(((φ ψ) (χ θ)) → τ)       (((φ χ) (θ ψ)) → τ)

Theoremanandi 505 Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
((φ (ψ χ)) ↔ ((φ ψ) (φ χ)))

Theoremanandir 506 Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
(((φ ψ) χ) ↔ ((φ χ) (ψ χ)))

Theoremanandis 507 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
(((φ ψ) (φ χ)) → τ)       ((φ (ψ χ)) → τ)

Theoremanandirs 508 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
(((φ χ) (ψ χ)) → τ)       (((φ ψ) χ) → τ)

Theoremimpbida 509 Deduce an equivalence from two implications. (Contributed by NM, 17-Feb-2007.)
((φ ψ) → χ)    &   ((φ χ) → ψ)       (φ → (ψχ))

Theorempm3.45 510 Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((φψ) → ((φ χ) → (ψ χ)))

Theoremim2anan9 511 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 512 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 513 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))

Theorempm5.1 514 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))

Theorempm3.43 515 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 516 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)))

Theorempm3.43OLD 517 Obsolete proof of pm3.43 515 as of 27-Nov-2013 (Contributed by NM, 3-Jan-2005.)
(((φψ) (φχ)) → (φ → (ψ χ)))

Theorempm4.76 518 Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
(((φψ) (φχ)) ↔ (φ → (ψ χ)))

Theorempm4.38 519 Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
(((φχ) (ψθ)) → ((φ ψ) ↔ (χ θ)))

Theorembi2anan9 520 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 521 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 522 Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)
(φ → (ψχ))    &   (θ → (τη))       ((φ θ) → ((ψτ) ↔ (χη)))

Theorempm5.33 523 Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((φ (ψχ)) ↔ (φ ((φ ψ) → χ)))

Theorempm5.36 524 Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((φ (φψ)) ↔ (ψ (φψ)))

Theorembianabs 525 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 526 'Not' introduction. Axiom 4 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
((φ → ¬ φ) → ¬ φ)

Axiomax-in2 527 'Not' elimination. Axiom 5 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
φ → (φψ))

Theorempm2.01 528 Reductio ad absurdum. Theorem *2.01 of [WhiteheadRussell] p. 100. This is valid intuitionistically (in the terminology of [Bauer] p. 482 it is a proof of negation not a proof by contradiction); compare with pm2.18 722 which is not. (Contributed by Mario Carneiro, 12-May-2015.)
((φ → ¬ φ) → ¬ φ)

Theorempm2.21 529 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 530 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 531 A contradiction implies anything. Deduction from pm2.21 529. (Contributed by NM, 10-Feb-1996.)
(φ → ¬ ψ)       (φ → (ψχ))

Theorempm2.24 532 Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
(φ → (¬ φψ))

Theorempm2.24d 533 Deduction version of pm2.24 532. (Contributed by NM, 30-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
(φψ)       (φ → (¬ ψχ))

Theorempm2.24i 534 Inference version of pm2.24 532. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
φ       φψ)

Theoremcon2d 535 A contraposition deduction. (Contributed by NM, 19-Aug-1993.) (Revised by NM, 12-Feb-2013.)
(φ → (ψ → ¬ χ))       (φ → (χ → ¬ ψ))

Theoremmt2d 536 Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
(φχ)    &   (φ → (ψ → ¬ χ))       (φ → ¬ ψ)

Theoremnsyl3 537 A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) (Revised by NM, 13-Jun-2013.)
(φ → ¬ ψ)    &   (χψ)       (χ → ¬ φ)

Theoremcon2i 538 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 539 A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
(φ → ¬ ψ)    &   (χψ)       (φ → ¬ χ)

Theoremnotnot1 540 Adding double negation. Theorem *2.12 of [WhiteheadRussell] p. 101. This one holds intuitionistically, but its converse, notnot2 725, does not. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
(φ → ¬ ¬ φ)

Theoremcon3d 541 A contraposition deduction. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)
(φ → (ψχ))       (φ → (¬ χ → ¬ ψ))

Theoremcon3i 542 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
(φψ)       ψ → ¬ φ)

Theoremcon3rr3 543 Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
(φ → (ψχ))       χ → (φ → ¬ ψ))

Theorempm3.2im 544 In classical logic, this is just a restatement of pm3.2 124. In intuitionistic logic, it still holds, but is weaker than pm3.2. (Contributed by Mario Carneiro, 12-May-2015.)
(φ → (ψ → ¬ (φ → ¬ ψ)))

Theoremexpi 545 An exportation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)
(¬ (φ → ¬ ψ) → χ)       (φ → (ψχ))

Theorempm2.65i 546 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 547 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 548 Theorem used to justify definition of intuitionistic biconditional df-bi 108. (Contributed by NM, 24-Nov-2017.)
((((φψ) (ψφ)) → ((φψ) (ψφ))) (((φψ) (ψφ)) → ((φψ) (ψφ))))

Theoremcon3 549 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 550 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 551 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
χ    &   (φ → (ψ → ¬ χ))       (φ → ¬ ψ)

Theoremnotnoti 552 Infer double negation. (Contributed by NM, 27-Feb-2008.)
φ        ¬ ¬ φ

Theorempm2.21i 553 A contradiction implies anything. Inference from pm2.21 529. (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ φ       (φψ)

Theorempm2.24ii 554 A contradiction implies anything. Inference from pm2.24 532. (Contributed by NM, 27-Feb-2008.)
φ    &    ¬ φ       ψ

Theoremnsyld 555 A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
(φ → (ψ → ¬ χ))    &   (φ → (τχ))       (φ → (ψ → ¬ τ))

Theoremnsyli 556 A negated syllogism inference. (Contributed by NM, 3-May-1994.)
(φ → (ψχ))    &   (θ → ¬ χ)       (φ → (θ → ¬ ψ))

Theoremmth8 557 Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
(φ → (¬ ψ → ¬ (φψ)))

Theoremjc 558 Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (φχ)       (φ → ¬ (ψ → ¬ χ))

Theorempm2.51 559 Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (φψ) → (φ → ¬ ψ))

Theorempm2.52 560 Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
(¬ (φψ) → (¬ φ → ¬ ψ))

Theoremexpt 561 Exportation theorem expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.)
((¬ (φ → ¬ ψ) → χ) → (φ → (ψχ)))

Theoremjarl 562 Elimination of a nested antecedent. Although it is a kind of reversal of inference ja 740 it holds intuitionistically, while ja 740 does not. (Contributed by Wolf Lammen, 10-May-2013.)
(((φψ) → χ) → (¬ φχ))

Theorempm2.65 563 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 [Bauer] p. 482). By contrast, proofs which assume ¬ φ, derive a contradiction, and conclude φ, such as condan 1854, are not valid intuitionistically. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.)
((φψ) → ((φ → ¬ ψ) → ¬ φ))

Theorempm2.65d 564 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 565 Deduction rule for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
((φ ψ) → χ)    &   ((φ ψ) → ¬ χ)       (φ → ¬ ψ)

Theoremmto 566 The rule of modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
¬ ψ    &   (φψ)        ¬ φ

Theoremmtod 567 Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
(φ → ¬ χ)    &   (φ → (ψχ))       (φ → ¬ ψ)

Theoremmtoi 568 Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
¬ χ    &   (φ → (ψχ))       (φ → ¬ ψ)

Theoremmtand 569 A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
(φ → ¬ χ)    &   ((φ ψ) → χ)       (φ → ¬ ψ)

Theoremnotbid 570 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 571 Contraposition. Bidirectional version of con2 550. (Contributed by NM, 5-Aug-1993.)
((φ → ¬ ψ) ↔ (ψ → ¬ φ))

Theoremnotbii 572 Negate both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
(φψ)       φ ↔ ¬ ψ)

Theoremmtbi 573 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 574 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 575 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
(φ → ¬ ψ)    &   (φ → (ψχ))       (φ → ¬ χ)

Theoremmtbird 576 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
(φ → ¬ χ)    &   (φ → (ψχ))       (φ → ¬ ψ)

Theoremmtbii 577 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
¬ ψ    &   (φ → (ψχ))       (φ → ¬ χ)

Theoremmtbiri 578 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
¬ χ    &   (φ → (ψχ))       (φ → ¬ ψ)

Theoremsylnib 579 A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
(φ → ¬ ψ)    &   (ψχ)       (φ → ¬ χ)

Theoremsylnibr 580 A mixed syllogism inference from an implication and a biconditional. Useful for substituting an consequent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.)
(φ → ¬ ψ)    &   (χψ)       (φ → ¬ χ)

Theoremsylnbi 581 A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.)
(φψ)    &   ψχ)       φχ)

Theoremsylnbir 582 A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
(ψφ)    &   ψχ)       φχ)

Theoremxchnxbi 583 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
φψ)    &   (φχ)       χψ)

Theoremxchnxbir 584 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
φψ)    &   (χφ)       χψ)

Theoremxchbinx 585 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(φ ↔ ¬ ψ)    &   (ψχ)       (φ ↔ ¬ χ)

Theoremxchbinxr 586 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(φ ↔ ¬ ψ)    &   (χψ)       (φ ↔ ¬ χ)

Theoremmt2bi 587 A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
φ       ψ ↔ (ψ → ¬ φ))

Theoremmtt 588 Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
φ → (¬ ψ ↔ (ψφ)))

Theorempm5.21 589 Two propositions are equivalent if they are both false. Theorem *5.21 of [WhiteheadRussell] p. 124. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
((¬ φ ¬ ψ) → (φψ))

Theorempm5.21im 590 Two propositions are equivalent if they are both false. Closed form of 2false 595. Equivalent to a bi2 119-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.) (Revised by Mario Carneiro, 31-Jan-2015.)
φ → (¬ ψ → (φψ)))

Theoremnbn2 591 The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
φ → (¬ ψ ↔ (φψ)))

Theorembibif 592 Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
ψ → ((φψ) ↔ ¬ φ))

Theoremnbn 593 The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
¬ φ       ψ ↔ (ψφ))

Theoremnbn3 594 Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
φ       ψ ↔ (ψ ↔ ¬ φ))

Theorem2false 595 Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ φ    &    ¬ ψ       (φψ)

Theorem2falsed 596 Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.)
(φ → ¬ ψ)    &   (φ → ¬ χ)       (φ → (ψχ))

Theorempm5.21ni 597 Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.)
(φψ)    &   (χψ)       ψ → (φχ))

Theorempm5.21nii 598 Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.) (Revised by Mario Carneiro, 31-Jan-2015.)
(φψ)    &   (χψ)    &   (ψ → (φχ))       (φχ)

Theorempm5.21ndd 599 Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Revised by Mario Carneiro, 31-Jan-2015.)
(φ → (χψ))    &   (φ → (θψ))    &   (φ → (ψ → (χθ)))       (φ → (χθ))

Theorempm5.19 600 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ (φ ↔ ¬ φ)

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