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Theorem List for Intuitionistic Logic Explorer - 601-700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmtand 601 A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
(𝜑 → ¬ 𝜒)    &   ((𝜑𝜓) → 𝜒)       (𝜑 → ¬ 𝜓)
 
Theoremnotbid 602 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 603 Contraposition. Bidirectional version of con2 582. (Contributed by NM, 5-Aug-1993.)
((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑))
 
Theoremnotbii 604 Negate both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑𝜓)       𝜑 ↔ ¬ 𝜓)
 
Theoremmtbi 605 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 606 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 607 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜒)
 
Theoremmtbird 608 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)
 
Theoremmtbii 609 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
¬ 𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜒)
 
Theoremmtbiri 610 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
¬ 𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → ¬ 𝜓)
 
Theoremsylnib 611 A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜓𝜒)       (𝜑 → ¬ 𝜒)
 
Theoremsylnibr 612 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 613 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 614 A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
(𝜓𝜑)    &   𝜓𝜒)       𝜑𝜒)
 
Theoremxchnxbi 615 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
𝜑𝜓)    &   (𝜑𝜒)       𝜒𝜓)
 
Theoremxchnxbir 616 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
𝜑𝜓)    &   (𝜒𝜑)       𝜒𝜓)
 
Theoremxchbinx 617 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(𝜑 ↔ ¬ 𝜓)    &   (𝜓𝜒)       (𝜑 ↔ ¬ 𝜒)
 
Theoremxchbinxr 618 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
(𝜑 ↔ ¬ 𝜓)    &   (𝜒𝜓)       (𝜑 ↔ ¬ 𝜒)
 
Theoremmt2bi 619 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 620 Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑 → (¬ 𝜓 ↔ (𝜓𝜑)))
 
Theorempm5.21 621 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 622 Two propositions are equivalent if they are both false. Closed form of 2false 627. Equivalent to a bi2 125-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑 → (¬ 𝜓 → (𝜑𝜓)))
 
Theoremnbn2 623 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 624 Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
𝜓 → ((𝜑𝜓) ↔ ¬ 𝜑))
 
Theoremnbn 625 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 626 Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
𝜑       𝜓 ↔ (𝜓 ↔ ¬ 𝜑))
 
Theorem2false 627 Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ 𝜑    &    ¬ 𝜓       (𝜑𝜓)
 
Theorem2falsed 628 Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → ¬ 𝜒)       (𝜑 → (𝜓𝜒))
 
Theorempm5.21ni 629 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 630 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 631 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))
 
Theorempm5.19 632 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ (𝜑 ↔ ¬ 𝜑)
 
Theorempm4.8 633 Theorem *4.8 of [WhiteheadRussell] p. 122. This one holds for all propositions, but compare with pm4.81dc 825 which requires a decidability condition. (Contributed by NM, 3-Jan-2005.)
((𝜑 → ¬ 𝜑) ↔ ¬ 𝜑)
 
Theoremimnan 634 Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.)
((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremimnani 635 Express implication in terms of conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
¬ (𝜑𝜓)       (𝜑 → ¬ 𝜓)
 
Theoremnan 636 Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
((𝜑 → ¬ (𝜓𝜒)) ↔ ((𝜑𝜓) → ¬ 𝜒))
 
Theorempm3.24 637 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)
 
Theoremnotnotnot 638 Triple negation. (Contributed by Jim Kingdon, 28-Jul-2018.)
(¬ ¬ ¬ 𝜑 ↔ ¬ 𝜑)
 
1.2.6  Logical disjunction
 
Syntaxwo 639 Extend wff definition to include disjunction ('or').
wff (𝜑𝜓)
 
Axiomax-io 640 Definition of 'or'. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias jaob 641 instead. (New usage is discouraged.)
(((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))
 
Theoremjaob 641 Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. Alias of ax-io 640. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))
 
Theoremolc 642 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 643 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))
 
Theorempm2.67-2 644 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))
 
Theorempm3.44 645 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 646 Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) (Revised by NM, 31-Jan-2015.)
(𝜑𝜓)    &   (𝜒𝜓)       ((𝜑𝜒) → 𝜓)
 
Theoremjaod 647 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 648 Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜓𝜃))       (𝜑𝜒)
 
Theoremjaao 649 Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))       ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))
 
Theoremjaoa 650 Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))       ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))
 
Theorempm2.53 651 Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 801). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.)
((𝜑𝜓) → (¬ 𝜑𝜓))
 
Theoremori 652 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑𝜓)       𝜑𝜓)
 
Theoremord 653 Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓𝜒))
 
Theoremorel1 654 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 655 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))
 
Theorempm1.4 656 Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 15-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremorcom 657 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))
 
Theoremorcomd 658 Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))
 
Theoremorcoms 659 Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
((𝜑𝜓) → 𝜒)       ((𝜓𝜑) → 𝜒)
 
Theoremorci 660 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑       (𝜑𝜓)
 
Theoremolci 661 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑       (𝜓𝜑)
 
Theoremorcd 662 Deduction introducing a disjunct. (Contributed by NM, 20-Sep-2007.)
(𝜑𝜓)       (𝜑 → (𝜓𝜒))
 
Theoremolcd 663 Deduction introducing a disjunct. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓))
 
Theoremorcs 664 Deduction eliminating disjunct. Notational convention: We sometimes suffix with "s" the label of an inference that manipulates an antecedent, leaving the consequent unchanged. The "s" means that the inference eliminates the need for a syllogism (syl 14) -type inference in a proof. (Contributed by NM, 21-Jun-1994.)
((𝜑𝜓) → 𝜒)       (𝜑𝜒)
 
Theoremolcs 665 Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
((𝜑𝜓) → 𝜒)       (𝜓𝜒)
 
Theorempm2.07 666 Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
(𝜑 → (𝜑𝜑))
 
Theorempm2.45 667 Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → ¬ 𝜑)
 
Theorempm2.46 668 Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → ¬ 𝜓)
 
Theorempm2.47 669 Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (¬ 𝜑𝜓))
 
Theorempm2.48 670 Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (𝜑 ∨ ¬ 𝜓))
 
Theorempm2.49 671 Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))
 
Theorempm2.67 672 Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
(((𝜑𝜓) → 𝜓) → (𝜑𝜓))
 
Theorembiorf 673 A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of [WhiteheadRussell] p. 121. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
𝜑 → (𝜓 ↔ (𝜑𝜓)))
 
Theorembiortn 674 A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
(𝜑 → (𝜓 ↔ (¬ 𝜑𝜓)))
 
Theorembiorfi 675 A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)
¬ 𝜑       (𝜓 ↔ (𝜓𝜑))
 
Theorempm2.621 676 Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 13-Dec-2013.)
((𝜑𝜓) → ((𝜑𝜓) → 𝜓))
 
Theorempm2.62 677 Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.)
((𝜑𝜓) → ((𝜑𝜓) → 𝜓))
 
Theoremimorri 678 Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑𝜓)       (𝜑𝜓)
 
Theoremioran 679 Negated disjunction in terms of conjunction. This version of DeMorgan's law is a biconditional for all propositions (not just decidable ones), unlike oranim 818, anordc 874, or ianordc 810. Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
(¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
 
Theorempm3.14 680 Theorem *3.14 of [WhiteheadRussell] p. 111. One direction of De Morgan's law). The biconditional holds for decidable propositions as seen at ianordc 810. The converse holds for decidable propositions, as seen at pm3.13dc 877. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑𝜓))
 
Theorempm3.1 681 Theorem *3.1 of [WhiteheadRussell] p. 111. The converse holds for decidable propositions, as seen at anordc 874. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
((𝜑𝜓) → ¬ (¬ 𝜑 ∨ ¬ 𝜓))
 
Theoremjao 682 Disjunction of antecedents. Compare Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 4-Apr-2013.)
((𝜑𝜓) → ((𝜒𝜓) → ((𝜑𝜒) → 𝜓)))
 
Theorempm1.2 683 Axiom *1.2 (Taut) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 10-Mar-2013.)
((𝜑𝜑) → 𝜑)
 
Theoremoridm 684 Idempotent law for disjunction. Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 16-Apr-2011.) (Proof shortened by Wolf Lammen, 10-Mar-2013.)
((𝜑𝜑) ↔ 𝜑)
 
Theorempm4.25 685 Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑𝜑))
 
Theoremorim12i 686 Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) → (𝜓𝜃))
 
Theoremorim1i 687 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
(𝜑𝜓)       ((𝜑𝜒) → (𝜓𝜒))
 
Theoremorim2i 688 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
(𝜑𝜓)       ((𝜒𝜑) → (𝜒𝜓))
 
Theoremorbi2i 689 Inference adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Dec-2012.)
(𝜑𝜓)       ((𝜒𝜑) ↔ (𝜒𝜓))
 
Theoremorbi1i 690 Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))
 
Theoremorbi12i 691 Infer the disjunction of two equivalences. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theorempm1.5 692 Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → (𝜓 ∨ (𝜑𝜒)))
 
Theoremor12 693 Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
((𝜑 ∨ (𝜓𝜒)) ↔ (𝜓 ∨ (𝜑𝜒)))
 
Theoremorass 694 Associative law for disjunction. Theorem *4.33 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((𝜑𝜓) ∨ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒)))
 
Theorempm2.31 695 Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → ((𝜑𝜓) ∨ 𝜒))
 
Theorempm2.32 696 Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∨ 𝜒) → (𝜑 ∨ (𝜓𝜒)))
 
Theoremor32 697 A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∨ 𝜓))
 
Theoremor4 698 Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∨ (𝜓𝜃)))
 
Theoremor42 699 Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∨ (𝜃𝜓)))
 
Theoremorordi 700 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒)))
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