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Theorem List for Intuitionistic Logic Explorer - 501-600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsimpllr 501 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜓)
 
Theoremsimplrl 502 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜓)
 
Theoremsimplrr 503 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜒)
 
Theoremsimprll 504 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜓)
 
Theoremsimprlr 505 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜒)
 
Theoremsimprrl 506 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜒)
 
Theoremsimprrr 507 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜃)
 
Theoremsimp-4l 508 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜑)
 
Theoremsimp-4r 509 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)
 
Theoremsimp-5l 510 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜑)
 
Theoremsimp-5r 511 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)
 
Theoremsimp-6l 512 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜑)
 
Theoremsimp-6r 513 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)
 
Theoremsimp-7l 514 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜑)
 
Theoremsimp-7r 515 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)
 
Theoremsimp-8l 516 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜑)
 
Theoremsimp-8r 517 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)
 
Theoremsimp-9l 518 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜑)
 
Theoremsimp-9r 519 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)
 
Theoremsimp-10l 520 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜑)
 
Theoremsimp-10r 521 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜓)
 
Theoremsimp-11l 522 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜑)
 
Theoremsimp-11r 523 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜓)
 
Theorempm4.87 524 Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.)
(((((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))) ∧ ((𝜑 → (𝜓𝜒)) ↔ (𝜓 → (𝜑𝜒)))) ∧ ((𝜓 → (𝜑𝜒)) ↔ ((𝜓𝜑) → 𝜒)))
 
Theoremabai 525 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.)
((𝜑𝜓) ↔ (𝜑 ∧ (𝜑𝜓)))
 
Theoreman12 526 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.)
((𝜑 ∧ (𝜓𝜒)) ↔ (𝜓 ∧ (𝜑𝜒)))
 
Theoreman32 527 A rearrangement of conjuncts. (Contributed by NM, 12-Mar-1995.) (Proof shortened by Wolf Lammen, 25-Dec-2012.)
(((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))
 
Theoreman13 528 A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
((𝜑 ∧ (𝜓𝜒)) ↔ (𝜒 ∧ (𝜓𝜑)))
 
Theoreman31 529 A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
(((𝜑𝜓) ∧ 𝜒) ↔ ((𝜒𝜓) ∧ 𝜑))
 
Theoreman12s 530 Swap two conjuncts in antecedent. The label suffix "s" means that an12 526 is combined with syl 14 (or a variant). (Contributed by NM, 13-Mar-1996.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜓 ∧ (𝜑𝜒)) → 𝜃)
 
Theoremancom2s 531 Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜒𝜓)) → 𝜃)
 
Theoreman13s 532 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜒 ∧ (𝜓𝜑)) → 𝜃)
 
Theoreman32s 533 Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜒) ∧ 𝜓) → 𝜃)
 
Theoremancom1s 534 Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜓𝜑) ∧ 𝜒) → 𝜃)
 
Theoreman31s 535 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜒𝜓) ∧ 𝜑) → 𝜃)
 
Theoremanass1rs 536 Commutative-associative law for conjunction in an antecedent. (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (((𝜑𝜒) ∧ 𝜓) → 𝜃)
 
Theoremanabs1 537 Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(((𝜑𝜓) ∧ 𝜑) ↔ (𝜑𝜓))
 
Theoremanabs5 538 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
((𝜑 ∧ (𝜑𝜓)) ↔ (𝜑𝜓))
 
Theoremanabs7 539 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.)
((𝜓 ∧ (𝜑𝜓)) ↔ (𝜑𝜓))
 
Theoremanabsan 540 Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) (Revised by NM, 18-Nov-2013.)
(((𝜑𝜑) ∧ 𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremanabss1 541 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
(((𝜑𝜓) ∧ 𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremanabss4 542 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
(((𝜓𝜑) ∧ 𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremanabss5 543 Absorption of antecedent into conjunction. (Contributed by NM, 10-May-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
((𝜑 ∧ (𝜑𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremanabsi5 544 Absorption of antecedent into conjunction. (Contributed by NM, 11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
(𝜑 → ((𝜑𝜓) → 𝜒))       ((𝜑𝜓) → 𝜒)
 
Theoremanabsi6 545 Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.)
(𝜑 → ((𝜓𝜑) → 𝜒))       ((𝜑𝜓) → 𝜒)
 
Theoremanabsi7 546 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
(𝜓 → ((𝜑𝜓) → 𝜒))       ((𝜑𝜓) → 𝜒)
 
Theoremanabsi8 547 Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.)
(𝜓 → ((𝜓𝜑) → 𝜒))       ((𝜑𝜓) → 𝜒)
 
Theoremanabss7 548 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 19-Nov-2013.)
((𝜓 ∧ (𝜑𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremanabsan2 549 Absorption of antecedent with conjunction. (Contributed by NM, 10-May-2004.) (Revised by NM, 1-Jan-2013.)
((𝜑 ∧ (𝜓𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremanabss3 550 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
(((𝜑𝜓) ∧ 𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoreman4 551 Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ (𝜓𝜃)))
 
Theoreman42 552 Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ (𝜃𝜓)))
 
Theoreman4s 553 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)
 
Theoreman42s 554 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       (((𝜑𝜒) ∧ (𝜃𝜓)) → 𝜏)
 
Theoremanandi 555 Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
 
Theoremanandir 556 Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
(((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))
 
Theoremanandis 557 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → 𝜏)       ((𝜑 ∧ (𝜓𝜒)) → 𝜏)
 
Theoremanandirs 558 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
(((𝜑𝜒) ∧ (𝜓𝜒)) → 𝜏)       (((𝜑𝜓) ∧ 𝜒) → 𝜏)
 
Theoremsyl2an2 559 syl2an 283 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
(𝜑𝜓)    &   ((𝜒𝜑) → 𝜃)    &   ((𝜓𝜃) → 𝜏)       ((𝜒𝜑) → 𝜏)
 
Theoremsyl2an2r 560 syl2anr 284 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
(𝜑𝜓)    &   ((𝜑𝜒) → 𝜃)    &   ((𝜓𝜃) → 𝜏)       ((𝜑𝜒) → 𝜏)
 
Theoremimpbida 561 Deduce an equivalence from two implications. (Contributed by NM, 17-Feb-2007.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜒) → 𝜓)       (𝜑 → (𝜓𝜒))
 
Theorempm3.45 562 Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜓𝜒)))
 
Theoremim2anan9 563 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜑𝜃) → ((𝜓𝜏) → (𝜒𝜂)))
 
Theoremim2anan9r 564 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜃𝜑) → ((𝜓𝜏) → (𝜒𝜂)))
 
Theoremanim12dan 565 Conjoin antecedents and consequents in a deduction. (Contributed by Mario Carneiro, 12-May-2014.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜏)       ((𝜑 ∧ (𝜓𝜃)) → (𝜒𝜏))
 
Theorempm5.1 566 Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.)
((𝜑𝜓) → (𝜑𝜓))
 
Theorempm3.43 567 Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 27-Nov-2013.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
 
Theoremjcab 568 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 → (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
 
Theorempm4.76 569 Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒)))
 
Theorempm4.38 570 Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜒) ∧ (𝜓𝜃)) → ((𝜑𝜓) ↔ (𝜒𝜃)))
 
Theorembi2anan9 571 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 31-Jul-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜑𝜃) → ((𝜓𝜏) ↔ (𝜒𝜂)))
 
Theorembi2anan9r 572 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜃𝜑) → ((𝜓𝜏) ↔ (𝜒𝜂)))
 
Theorembi2bian9 573 Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜑𝜃) → ((𝜓𝜏) ↔ (𝜒𝜂)))
 
Theorempm5.33 574 Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∧ (𝜓𝜒)) ↔ (𝜑 ∧ ((𝜑𝜓) → 𝜒)))
 
Theorempm5.36 575 Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∧ (𝜑𝜓)) ↔ (𝜓 ∧ (𝜑𝜓)))
 
Theorembianabs 576 Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.)
(𝜑 → (𝜓 ↔ (𝜑𝜒)))       (𝜑 → (𝜓𝜒))
 
1.2.5  Logical negation (intuitionistic)
 
Axiomax-in1 577 'Not' introduction. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
((𝜑 → ¬ 𝜑) → ¬ 𝜑)
 
Axiomax-in2 578 'Not' elimination. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
𝜑 → (𝜑𝜓))
 
Theorempm2.01 579 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 786 which only holds for some propositions. (Contributed by Mario Carneiro, 12-May-2015.)
((𝜑 → ¬ 𝜑) → ¬ 𝜑)
 
Theorempm2.21 580 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 581 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 582 A contradiction implies anything. Deduction from pm2.21 580. (Contributed by NM, 10-Feb-1996.)
(𝜑 → ¬ 𝜓)       (𝜑 → (𝜓𝜒))
 
Theorempm2.21dd 583 A contradiction implies anything. Deduction from pm2.21 580. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)       (𝜑𝜒)
 
Theorempm2.24 584 Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
(𝜑 → (¬ 𝜑𝜓))
 
Theorempm2.24d 585 Deduction version of pm2.24 584. (Contributed by NM, 30-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑𝜓)       (𝜑 → (¬ 𝜓𝜒))
 
Theorempm2.24i 586 Inference version of pm2.24 584. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑       𝜑𝜓)
 
Theoremcon2d 587 A contraposition deduction. (Contributed by NM, 19-Aug-1993.) (Revised by NM, 12-Feb-2013.)
(𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → (𝜒 → ¬ 𝜓))
 
Theoremmt2d 588 Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
(𝜑𝜒)    &   (𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → ¬ 𝜓)
 
Theoremnsyl3 589 A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) (Revised by NM, 13-Jun-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜒𝜓)       (𝜒 → ¬ 𝜑)
 
Theoremcon2i 590 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 591 A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜒𝜓)       (𝜑 → ¬ 𝜒)
 
Theoremnotnot 592 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 787). (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
(𝜑 → ¬ ¬ 𝜑)
 
Theoremnotnotd 593 Deduction associated with notnot 592 and notnoti 607. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
(𝜑𝜓)       (𝜑 → ¬ ¬ 𝜓)
 
Theoremcon3d 594 A contraposition deduction. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜒 → ¬ 𝜓))
 
Theoremcon3i 595 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
(𝜑𝜓)       𝜓 → ¬ 𝜑)
 
Theoremcon3rr3 596 Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
(𝜑 → (𝜓𝜒))       𝜒 → (𝜑 → ¬ 𝜓))
 
Theoremcon3dimp 597 Variant of con3d 594 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 → (𝜓𝜒))       ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)
 
Theorempm2.01da 598 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑𝜓) → ¬ 𝜓)       (𝜑 → ¬ 𝜓)
 
Theorempm3.2im 599 In classical logic, this is just a restatement of pm3.2 137. In intuitionistic logic, it still holds, but is weaker than pm3.2. (Contributed by Mario Carneiro, 12-May-2015.)
(𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓)))
 
Theoremexpi 600 An exportation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)
(¬ (𝜑 → ¬ 𝜓) → 𝜒)       (𝜑 → (𝜓𝜒))
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