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

Theoremsimpll 501 Simplification of a conjunction. (Contributed by NM, 18-Mar-2007.)
(((𝜑𝜓) ∧ 𝜒) → 𝜑)

Theoremsimplr 502 Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.)
(((𝜑𝜓) ∧ 𝜒) → 𝜓)

Theoremsimprl 503 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
((𝜑 ∧ (𝜓𝜒)) → 𝜓)

Theoremsimprr 504 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
((𝜑 ∧ (𝜓𝜒)) → 𝜒)

Theoremsimplll 505 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜑)

Theoremsimpllr 506 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜓)

Theoremsimplrl 507 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜓)

Theoremsimplrr 508 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜒)

Theoremsimprll 509 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜓)

Theoremsimprlr 510 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜒)

Theoremsimprrl 511 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜒)

Theoremsimprrr 512 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜃)

Theoremsimp-4l 513 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜑)

Theoremsimp-4r 514 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)

Theoremsimp-5l 515 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜑)

Theoremsimp-5r 516 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)

Theoremsimp-6l 517 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜑)

Theoremsimp-6r 518 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)

Theoremsimp-7l 519 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜑)

Theoremsimp-7r 520 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)

Theoremsimp-8l 521 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜑)

Theoremsimp-8r 522 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)

Theoremsimp-9l 523 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜑)

Theoremsimp-9r 524 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)

Theoremsimp-10l 525 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜑)

Theoremsimp-10r 526 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
(((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜓)

Theoremsimp-11l 527 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 528 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
((((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜓)

Theorempm4.87 529 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)))

Theorema2and 530 Deduction distributing a conjunction as embedded antecedent. (Contributed by AV, 25-Oct-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)
(𝜑 → ((𝜓𝜌) → (𝜏𝜃)))    &   (𝜑 → ((𝜓𝜌) → 𝜒))       (𝜑 → (((𝜓𝜒) → 𝜏) → ((𝜓𝜌) → 𝜃)))

Theoremanimpimp2impd 531 Deduction deriving nested implications from conjunctions. (Contributed by AV, 21-Aug-2022.)
((𝜓𝜑) → (𝜒 → (𝜃𝜂)))    &   ((𝜓 ∧ (𝜑𝜃)) → (𝜂𝜏))       (𝜑 → ((𝜓𝜒) → (𝜓 → (𝜃𝜏))))

Theoremabai 532 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 533 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 534 A rearrangement of conjuncts. (Contributed by NM, 12-Mar-1995.) (Proof shortened by Wolf Lammen, 25-Dec-2012.)
(((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))

Theoreman13 535 A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
((𝜑 ∧ (𝜓𝜒)) ↔ (𝜒 ∧ (𝜓𝜑)))

Theoreman31 536 A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
(((𝜑𝜓) ∧ 𝜒) ↔ ((𝜒𝜓) ∧ 𝜑))

Theoreman12s 537 Swap two conjuncts in antecedent. The label suffix "s" means that an12 533 is combined with syl 14 (or a variant). (Contributed by NM, 13-Mar-1996.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜓 ∧ (𝜑𝜒)) → 𝜃)

Theoremancom2s 538 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 539 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜒 ∧ (𝜓𝜑)) → 𝜃)

Theoreman32s 540 Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜒) ∧ 𝜓) → 𝜃)

Theoremancom1s 541 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 542 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜒𝜓) ∧ 𝜑) → 𝜃)

Theoremanass1rs 543 Commutative-associative law for conjunction in an antecedent. (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (((𝜑𝜒) ∧ 𝜓) → 𝜃)

Theoremanabs1 544 Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(((𝜑𝜓) ∧ 𝜑) ↔ (𝜑𝜓))

Theoremanabs5 545 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
((𝜑 ∧ (𝜑𝜓)) ↔ (𝜑𝜓))

Theoremanabs7 546 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.)
((𝜓 ∧ (𝜑𝜓)) ↔ (𝜑𝜓))

Theoremanabsan 547 Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) (Revised by NM, 18-Nov-2013.)
(((𝜑𝜑) ∧ 𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremanabss1 548 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 549 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
(((𝜓𝜑) ∧ 𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremanabss5 550 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 551 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 552 Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.)
(𝜑 → ((𝜓𝜑) → 𝜒))       ((𝜑𝜓) → 𝜒)

Theoremanabsi7 553 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 554 Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.)
(𝜓 → ((𝜓𝜑) → 𝜒))       ((𝜑𝜓) → 𝜒)

Theoremanabss7 555 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 556 Absorption of antecedent with conjunction. (Contributed by NM, 10-May-2004.) (Revised by NM, 1-Jan-2013.)
((𝜑 ∧ (𝜓𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremanabss3 557 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 558 Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ (𝜓𝜃)))

Theoreman42 559 Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ (𝜃𝜓)))

Theoreman4s 560 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)

Theoreman42s 561 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       (((𝜑𝜒) ∧ (𝜃𝜓)) → 𝜏)

Theoremanandi 562 Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Theoremanandir 563 Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
(((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))

Theoremanandis 564 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → 𝜏)       ((𝜑 ∧ (𝜓𝜒)) → 𝜏)

Theoremanandirs 565 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
(((𝜑𝜒) ∧ (𝜓𝜒)) → 𝜏)       (((𝜑𝜓) ∧ 𝜒) → 𝜏)

Theoremsyl2an2 566 syl2an 285 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
(𝜑𝜓)    &   ((𝜒𝜑) → 𝜃)    &   ((𝜓𝜃) → 𝜏)       ((𝜒𝜑) → 𝜏)

Theoremsyl2an2r 567 syl2anr 286 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
(𝜑𝜓)    &   ((𝜑𝜒) → 𝜃)    &   ((𝜓𝜃) → 𝜏)       ((𝜑𝜒) → 𝜏)

Theoremimpbida 568 Deduce an equivalence from two implications. (Contributed by NM, 17-Feb-2007.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜒) → 𝜓)       (𝜑 → (𝜓𝜒))

Theorempm3.45 569 Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜓𝜒)))

Theoremim2anan9 570 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜑𝜃) → ((𝜓𝜏) → (𝜒𝜂)))

Theoremim2anan9r 571 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜃𝜑) → ((𝜓𝜏) → (𝜒𝜂)))

Theoremanim12dan 572 Conjoin antecedents and consequents in a deduction. (Contributed by Mario Carneiro, 12-May-2014.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜏)       ((𝜑 ∧ (𝜓𝜃)) → (𝜒𝜏))

Theorempm5.1 573 Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.)
((𝜑𝜓) → (𝜑𝜓))

Theorempm3.43 574 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 575 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)))

Theorempm4.76 576 Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒)))

Theorempm4.38 577 Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜒) ∧ (𝜓𝜃)) → ((𝜑𝜓) ↔ (𝜒𝜃)))

Theorembi2anan9 578 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 31-Jul-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜑𝜃) → ((𝜓𝜏) ↔ (𝜒𝜂)))

Theorembi2anan9r 579 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜃𝜑) → ((𝜓𝜏) ↔ (𝜒𝜂)))

Theorembi2bian9 580 Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜑𝜃) → ((𝜓𝜏) ↔ (𝜒𝜂)))

Theorempm5.33 581 Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∧ (𝜓𝜒)) ↔ (𝜑 ∧ ((𝜑𝜓) → 𝜒)))

Theorempm5.36 582 Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∧ (𝜑𝜓)) ↔ (𝜓 ∧ (𝜑𝜓)))

Theorembianabs 583 Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.)
(𝜑 → (𝜓 ↔ (𝜑𝜒)))       (𝜑 → (𝜓𝜒))

Theorembiadani 584 An implication implies to the equivalence of some implied equivalence and some other equivalence involving a conjunction. (Contributed by BJ, 4-Mar-2023.)
(𝜑𝜓)       ((𝜓 → (𝜑𝜒)) ↔ (𝜑 ↔ (𝜓𝜒)))

Theorembiadanii 585 Inference associated with biadani 584. Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.) (Proof shortened by BJ, 4-Mar-2023.)
(𝜑𝜓)    &   (𝜓 → (𝜑𝜒))       (𝜑 ↔ (𝜓𝜒))

1.2.5  Logical negation (intuitionistic)

Axiomax-in1 586 'Not' introduction. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
((𝜑 → ¬ 𝜑) → ¬ 𝜑)

Axiomax-in2 587 'Not' elimination. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
𝜑 → (𝜑𝜓))

Theorempm2.01 588 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 823 which only holds for some propositions. (Contributed by Mario Carneiro, 12-May-2015.)
((𝜑 → ¬ 𝜑) → ¬ 𝜑)

Theorempm2.21 589 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 590 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 591 A contradiction implies anything. Deduction from pm2.21 589. (Contributed by NM, 10-Feb-1996.)
(𝜑 → ¬ 𝜓)       (𝜑 → (𝜓𝜒))

Theorempm2.21dd 592 A contradiction implies anything. Deduction from pm2.21 589. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)       (𝜑𝜒)

Theorempm2.24 593 Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
(𝜑 → (¬ 𝜑𝜓))

Theorempm2.24d 594 Deduction version of pm2.24 593. (Contributed by NM, 30-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑𝜓)       (𝜑 → (¬ 𝜓𝜒))

Theorempm2.24i 595 Inference version of pm2.24 593. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑       𝜑𝜓)

Theoremcon2d 596 A contraposition deduction. (Contributed by NM, 19-Aug-1993.) (Revised by NM, 12-Feb-2013.)
(𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → (𝜒 → ¬ 𝜓))

Theoremmt2d 597 Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
(𝜑𝜒)    &   (𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → ¬ 𝜓)

Theoremnsyl3 598 A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) (Revised by NM, 13-Jun-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜒𝜓)       (𝜒 → ¬ 𝜑)

Theoremcon2i 599 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 600 A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜒𝜓)       (𝜑 → ¬ 𝜒)

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