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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 ∧ (𝜓𝜒)) → 𝜃)       ((𝜒 ∧ (𝜓𝜑)) → 𝜃)
 
Theoreman32s 502 Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜒) ∧ 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.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜓𝜑) ∧ 𝜒) → 𝜃)
 
Theoreman31s 504 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜒𝜓) ∧ 𝜑) → 𝜃)
 
Theoremanass1rs 505 Commutative-associative law for conjunction in an antecedent. (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (((𝜑𝜒) ∧ 𝜓) → 𝜃)
 
Theoremanabs1 506 Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
(((𝜑𝜓) ∧ 𝜑) ↔ (𝜑𝜓))
 
Theoremanabs5 507 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
((𝜑 ∧ (𝜑𝜓)) ↔ (𝜑𝜓))
 
Theoremanabs7 508 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.)
((𝜓 ∧ (𝜑𝜓)) ↔ (𝜑𝜓))
 
Theoremanabsan 509 Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) (Revised by NM, 18-Nov-2013.)
(((𝜑𝜑) ∧ 𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremanabss1 510 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 511 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
(((𝜓𝜑) ∧ 𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremanabss5 512 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 513 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 514 Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.)
(𝜑 → ((𝜓𝜑) → 𝜒))       ((𝜑𝜓) → 𝜒)
 
Theoremanabsi7 515 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 516 Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.)
(𝜓 → ((𝜓𝜑) → 𝜒))       ((𝜑𝜓) → 𝜒)
 
Theoremanabss7 517 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 518 Absorption of antecedent with conjunction. (Contributed by NM, 10-May-2004.) (Revised by NM, 1-Jan-2013.)
((𝜑 ∧ (𝜓𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremanabss3 519 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 520 Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ (𝜓𝜃)))
 
Theoreman42 521 Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ (𝜃𝜓)))
 
Theoreman4s 522 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)
 
Theoreman42s 523 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       (((𝜑𝜒) ∧ (𝜃𝜓)) → 𝜏)
 
Theoremanandi 524 Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
 
Theoremanandir 525 Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
(((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜒) ∧ (𝜓𝜒)))
 
Theoremanandis 526 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → 𝜏)       ((𝜑 ∧ (𝜓𝜒)) → 𝜏)
 
Theoremanandirs 527 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
(((𝜑𝜒) ∧ (𝜓𝜒)) → 𝜏)       (((𝜑𝜓) ∧ 𝜒) → 𝜏)
 
Theoremimpbida 528 Deduce an equivalence from two implications. (Contributed by NM, 17-Feb-2007.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜒) → 𝜓)       (𝜑 → (𝜓𝜒))
 
Theorempm3.45 529 Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜓𝜒)))
 
Theoremim2anan9 530 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜑𝜃) → ((𝜓𝜏) → (𝜒𝜂)))
 
Theoremim2anan9r 531 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜃𝜑) → ((𝜓𝜏) → (𝜒𝜂)))
 
Theoremanim12dan 532 Conjoin antecedents and consequents in a deduction. (Contributed by Mario Carneiro, 12-May-2014.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜏)       ((𝜑 ∧ (𝜓𝜃)) → (𝜒𝜏))
 
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.)
((𝜑𝜓) → (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 → (𝜓𝜒)))
 
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 → (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
 
Theorempm4.76 536 Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒)))
 
Theorempm4.38 537 Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜒) ∧ (𝜓𝜃)) → ((𝜑𝜓) ↔ (𝜒𝜃)))
 
Theorembi2anan9 538 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 31-Jul-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜑𝜃) → ((𝜓𝜏) ↔ (𝜒𝜂)))
 
Theorembi2anan9r 539 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜃𝜑) → ((𝜓𝜏) ↔ (𝜒𝜂)))
 
Theorembi2bian9 540 Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜂))       ((𝜑𝜃) → ((𝜓𝜏) ↔ (𝜒𝜂)))
 
Theorempm5.33 541 Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∧ (𝜓𝜒)) ↔ (𝜑 ∧ ((𝜑𝜓) → 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 ↔ (𝜑𝜒)))       (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.)
((𝜑𝜓) → ¬ 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.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜓) → ¬ 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)    &   ((𝜑𝜓) → 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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