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Theorem List for Intuitionistic Logic Explorer - 601-700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnotnot 601 Double negation introduction. Theorem *2.12 of [WhiteheadRussell] p. 101. The converse need not hold. It holds exactly for stable propositions (by definition, see df-stab 799) and in particular for decidable propositions (see notnotrdc 811). See also notnotnot 606. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
(𝜑 → ¬ ¬ 𝜑)

Theoremnotnotd 602 Deduction associated with notnot 601 and notnoti 617. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
(𝜑𝜓)       (𝜑 → ¬ ¬ 𝜓)

Theoremcon3d 603 A contraposition deduction. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜒 → ¬ 𝜓))

Theoremcon3i 604 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
(𝜑𝜓)       𝜓 → ¬ 𝜑)

Theoremcon3rr3 605 Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
(𝜑 → (𝜓𝜒))       𝜒 → (𝜑 → ¬ 𝜓))

Theoremnotnotnot 606 Triple negation is equivalent to negation. (Contributed by Jim Kingdon, 28-Jul-2018.)
(¬ ¬ ¬ 𝜑 ↔ ¬ 𝜑)

Theoremcon3dimp 607 Variant of con3d 603 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 → (𝜓𝜒))       ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)

Theorempm2.01da 608 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
((𝜑𝜓) → ¬ 𝜓)       (𝜑 → ¬ 𝜓)

Theorempm3.2im 609 In classical logic, this is just a restatement of pm3.2 138. In intuitionistic logic, it still holds, but is weaker than pm3.2. (Contributed by Mario Carneiro, 12-May-2015.)
(𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓)))

Theoremexpi 610 An exportation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)
(¬ (𝜑 → ¬ 𝜓) → 𝜒)       (𝜑 → (𝜓𝜒))

Theorempm2.65i 611 Inference for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜓)        ¬ 𝜑

Theoremmt2 612 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 613 Theorem used to justify definition of intuitionistic biconditional df-bi 116. (Contributed by NM, 24-Nov-2017.)
((((𝜑𝜓) ∧ (𝜓𝜑)) → ((𝜑𝜓) ∧ (𝜓𝜑))) ∧ (((𝜑𝜓) ∧ (𝜓𝜑)) → ((𝜑𝜓) ∧ (𝜓𝜑))))

Theoremcon3 614 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 615 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 616 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
𝜒    &   (𝜑 → (𝜓 → ¬ 𝜒))       (𝜑 → ¬ 𝜓)

Theoremnotnoti 617 Infer double negation. (Contributed by NM, 27-Feb-2008.)
𝜑        ¬ ¬ 𝜑

Theorempm2.21i 618 A contradiction implies anything. Inference from pm2.21 589. (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ 𝜑       (𝜑𝜓)

Theorempm2.24ii 619 A contradiction implies anything. Inference from pm2.24 593. (Contributed by NM, 27-Feb-2008.)
𝜑    &    ¬ 𝜑       𝜓

Theoremnsyld 620 A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
(𝜑 → (𝜓 → ¬ 𝜒))    &   (𝜑 → (𝜏𝜒))       (𝜑 → (𝜓 → ¬ 𝜏))

Theoremnsyli 621 A negated syllogism inference. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → ¬ 𝜒)       (𝜑 → (𝜃 → ¬ 𝜓))

Theoremmth8 622 Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
(𝜑 → (¬ 𝜓 → ¬ (𝜑𝜓)))

Theoremjc 623 Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)    &   (𝜑𝜒)       (𝜑 → ¬ (𝜓 → ¬ 𝜒))

Theoremjcn 624 Inference joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜒)       (𝜑 → ¬ (𝜓𝜒))

Theoremconax1 625 Contrapositive of ax-1 6. (Contributed by BJ, 28-Oct-2023.)
(¬ (𝜑𝜓) → ¬ 𝜓)

Theoremconax1k 626 Weakening of conax1 625. General instance of pm2.51 627 and of pm2.52 628. (Contributed by BJ, 28-Oct-2023.)
(¬ (𝜑𝜓) → (𝜒 → ¬ 𝜓))

Theorempm2.51 627 Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (𝜑 → ¬ 𝜓))

Theorempm2.52 628 Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
(¬ (𝜑𝜓) → (¬ 𝜑 → ¬ 𝜓))

Theoremexpt 629 Exportation theorem pm3.3 259 (closed form of ex 114) expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.)
((¬ (𝜑 → ¬ 𝜓) → 𝜒) → (𝜑 → (𝜓𝜒)))

Theoremjarl 630 Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.)
(((𝜑𝜓) → 𝜒) → (¬ 𝜑𝜒))

Theorempm2.65 631 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 849, 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 632 Deduction 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 633 Deduction for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜓) → ¬ 𝜒)       (𝜑 → ¬ 𝜓)

Theoremmto 634 The rule of modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
¬ 𝜓    &   (𝜑𝜓)        ¬ 𝜑

Theoremmtod 635 Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)

Theoremmtoi 636 Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
¬ 𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)

Theoremmtand 637 A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
(𝜑 → ¬ 𝜒)    &   ((𝜑𝜓) → 𝜒)       (𝜑 → ¬ 𝜓)

Theoremnotbi 638 Equivalence property for negation. Closed form. (Contributed by BJ, 27-Jan-2020.)
((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))

Theoremnotbid 639 Equivalence property for negation. Deduction form. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))

Theoremnotbii 640 Equivalence property for negation. Inference form. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑𝜓)       𝜑 ↔ ¬ 𝜓)

Theoremcon2b 641 Contraposition. Bidirectional version of con2 615. (Contributed by NM, 5-Aug-1993.)
((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑))

Theoremmtbi 642 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 643 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 644 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜒)

Theoremmtbird 645 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)

Theoremmtbii 646 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
¬ 𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜒)

Theoremmtbiri 647 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
¬ 𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)

Theoremsylnib 648 A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜓𝜒)       (𝜑 → ¬ 𝜒)

Theoremsylnibr 649 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 650 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 651 A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜓𝜑)    &   𝜓𝜒)       𝜑𝜒)

Theoremxchnxbi 652 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
𝜑𝜓)    &   (𝜑𝜒)       𝜒𝜓)

Theoremxchnxbir 653 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
𝜑𝜓)    &   (𝜒𝜑)       𝜒𝜓)

Theoremxchbinx 654 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(𝜑 ↔ ¬ 𝜓)    &   (𝜓𝜒)       (𝜑 ↔ ¬ 𝜒)

Theoremxchbinxr 655 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(𝜑 ↔ ¬ 𝜓)    &   (𝜒𝜓)       (𝜑 ↔ ¬ 𝜒)

Theoremmt2bi 656 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 657 Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑 → (¬ 𝜓 ↔ (𝜓𝜑)))

Theoremannimim 658 Express conjunction in terms of implication. One direction of Theorem *4.61 of [WhiteheadRussell] p. 120. The converse holds for decidable propositions, as can be seen at annimdc 904. (Contributed by Jim Kingdon, 24-Dec-2017.)
((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))

Theorempm4.65r 659 One direction of Theorem *4.65 of [WhiteheadRussell] p. 120. The converse holds in classical logic. (Contributed by Jim Kingdon, 28-Jul-2018.)
((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (¬ 𝜑𝜓))

Theoremimanim 660 Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 857. (Contributed by Jim Kingdon, 24-Dec-2017.)
((𝜑𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))

Theorempm3.37 661 Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) → 𝜒) → ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))

Theoremimnan 662 Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.)
((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))

Theoremimnani 663 Express implication in terms of conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
¬ (𝜑𝜓)       (𝜑 → ¬ 𝜓)

Theoremnan 664 Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
((𝜑 → ¬ (𝜓𝜒)) ↔ ((𝜑𝜓) → ¬ 𝜒))

Theorempm3.24 665 Law of noncontradiction. Theorem *3.24 of [WhiteheadRussell] p. 111 (who call it the "law of contradiction"). (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 2-Feb-2015.)
¬ (𝜑 ∧ ¬ 𝜑)

Theorempm4.15 666 Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
(((𝜑𝜓) → ¬ 𝜒) ↔ ((𝜓𝜒) → ¬ 𝜑))

Theorempm5.21 667 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 668 Two propositions are equivalent if they are both false. Closed form of 2false 673. Equivalent to a bi2 129-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑 → (¬ 𝜓 → (𝜑𝜓)))

Theoremnbn2 669 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 670 Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))

Theoremnbn 671 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 672 Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
𝜑       𝜓 ↔ (𝜓 ↔ ¬ 𝜑))

Theorem2false 673 Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ 𝜑    &    ¬ 𝜓       (𝜑𝜓)

Theorem2falsed 674 Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → ¬ 𝜒)       (𝜑 → (𝜓𝜒))

Theorempm5.21ni 675 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 676 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 677 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))

Theorempm5.19 678 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ (𝜑 ↔ ¬ 𝜑)

Theorempm4.8 679 Theorem *4.8 of [WhiteheadRussell] p. 122. This one holds for all propositions, but compare with pm4.81dc 876 which requires a decidability condition. (Contributed by NM, 3-Jan-2005.)
((𝜑 → ¬ 𝜑) ↔ ¬ 𝜑)

1.2.6  Logical disjunction

Syntaxwo 680 Extend wff definition to include disjunction ('or').
wff (𝜑𝜓)

Axiomax-io 681 Definition of 'or'. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias jaob 682 instead. (New usage is discouraged.)
(((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))

Theoremjaob 682 Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. Alias of ax-io 681. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))

Theoremolc 683 Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.)
(𝜑 → (𝜓𝜑))

Theoremorc 684 Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.)
(𝜑 → (𝜑𝜓))

Theorempm2.67-2 685 Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
(((𝜑𝜒) → 𝜓) → (𝜑𝜓))

Theoremoibabs 686 Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)
(((𝜑𝜓) → (𝜑𝜓)) ↔ (𝜑𝜓))

Theorempm3.44 687 Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(((𝜓𝜑) ∧ (𝜒𝜑)) → ((𝜓𝜒) → 𝜑))

Theoremjaoi 688 Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) (Revised by NM, 31-Jan-2015.)
(𝜑𝜓)    &   (𝜒𝜓)       ((𝜑𝜒) → 𝜓)

Theoremjaod 689 Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.) (Revised by NM, 4-Apr-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))       (𝜑 → ((𝜓𝜃) → 𝜒))

Theoremmpjaod 690 Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜓𝜃))       (𝜑𝜒)

Theoremjaao 691 Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))       ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))

Theoremjaoa 692 Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))       ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))

Theoremimorr 693 Implication in terms of disjunction. One direction of theorem *4.6 of [WhiteheadRussell] p. 120. The converse holds for decidable propositions, as seen at imordc 865. (Contributed by Jim Kingdon, 21-Jul-2018.)
((¬ 𝜑𝜓) → (𝜑𝜓))

Theorempm2.53 694 Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 859). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.)
((𝜑𝜓) → (¬ 𝜑𝜓))

Theoremori 695 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑𝜓)       𝜑𝜓)

Theoremord 696 Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓𝜒))

Theoremorel1 697 Elimination of disjunction by denial of a disjunct. Theorem *2.55 of [WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Wolf Lammen, 21-Jul-2012.)
𝜑 → ((𝜑𝜓) → 𝜓))

Theoremorel2 698 Elimination of disjunction by denial of a disjunct. Theorem *2.56 of [WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Wolf Lammen, 5-Apr-2013.)
𝜑 → ((𝜓𝜑) → 𝜓))

Theorempm1.4 699 Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 15-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))

Theoremorcom 700 Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Nov-2012.)
((𝜑𝜓) ↔ (𝜓𝜑))

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