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Theorem List for Intuitionistic Logic Explorer - 701-800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempm5.19 701 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
¬ (𝜑 ↔ ¬ 𝜑)
 
Theorempm4.8 702 Theorem *4.8 of [WhiteheadRussell] p. 122. This one holds for all propositions, but compare with pm4.81dc 903 which requires a decidability condition. (Contributed by NM, 3-Jan-2005.)
((𝜑 → ¬ 𝜑) ↔ ¬ 𝜑)
 
1.2.6  Logical disjunction
 
Syntaxwo 703 Extend wff definition to include disjunction ('or').
wff (𝜑𝜓)
 
Axiomax-io 704 Definition of 'or'. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias jaob 705 instead. (New usage is discouraged.)
(((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))
 
Theoremjaob 705 Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. Alias of ax-io 704. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))
 
Theoremolc 706 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 707 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))
 
Theorempm2.67-2 708 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 709 Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)
(((𝜑𝜓) → (𝜑𝜓)) ↔ (𝜑𝜓))
 
Theorempm3.44 710 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 711 Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) (Revised by NM, 31-Jan-2015.)
(𝜑𝜓)    &   (𝜒𝜓)       ((𝜑𝜒) → 𝜓)
 
Theoremjaod 712 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 713 Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜓𝜃))       (𝜑𝜒)
 
Theoremjaao 714 Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))       ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))
 
Theoremjaoa 715 Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))       ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))
 
Theoremimorr 716 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 892. (Contributed by Jim Kingdon, 21-Jul-2018.)
((¬ 𝜑𝜓) → (𝜑𝜓))
 
Theorempm2.53 717 Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 886). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.)
((𝜑𝜓) → (¬ 𝜑𝜓))
 
Theoremori 718 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑𝜓)       𝜑𝜓)
 
Theoremord 719 Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑 → (𝜓𝜒))       (𝜑 → (¬ 𝜓𝜒))
 
Theoremorel1 720 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 721 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))
 
Theorempm1.4 722 Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 15-Nov-2012.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremorcom 723 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 724 Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜒𝜓))
 
Theoremorcoms 725 Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
((𝜑𝜓) → 𝜒)       ((𝜓𝜑) → 𝜒)
 
Theoremorci 726 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑       (𝜑𝜓)
 
Theoremolci 727 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑       (𝜓𝜑)
 
Theoremorcd 728 Deduction introducing a disjunct. (Contributed by NM, 20-Sep-2007.)
(𝜑𝜓)       (𝜑 → (𝜓𝜒))
 
Theoremolcd 729 Deduction introducing a disjunct. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓))
 
Theoremorcs 730 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 731 Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
((𝜑𝜓) → 𝜒)       (𝜓𝜒)
 
Theorempm2.07 732 Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
(𝜑 → (𝜑𝜑))
 
Theorempm2.45 733 Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → ¬ 𝜑)
 
Theorempm2.46 734 Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → ¬ 𝜓)
 
Theorempm2.47 735 Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (¬ 𝜑𝜓))
 
Theorempm2.48 736 Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (𝜑 ∨ ¬ 𝜓))
 
Theorempm2.49 737 Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))
 
Theorempm2.67 738 Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
(((𝜑𝜓) → 𝜓) → (𝜑𝜓))
 
Theorembiorf 739 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 740 A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
(𝜑 → (𝜓 ↔ (¬ 𝜑𝜓)))
 
Theorembiorfi 741 A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)
¬ 𝜑       (𝜓 ↔ (𝜓𝜑))
 
Theorempm2.621 742 Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 13-Dec-2013.)
((𝜑𝜓) → ((𝜑𝜓) → 𝜓))
 
Theorempm2.62 743 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 744 Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 31-Jan-2015.)
𝜑𝜓)       (𝜑𝜓)
 
Theorempm4.52im 745 One direction of theorem *4.52 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.)
((𝜑 ∧ ¬ 𝜓) → ¬ (¬ 𝜑𝜓))
 
Theorempm4.53r 746 One direction of theorem *4.53 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.)
((¬ 𝜑𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))
 
Theoremioran 747 Negated disjunction in terms of conjunction. This version of DeMorgan's law is a biconditional for all propositions (not just decidable ones), unlike oranim 776, anordc 951, or ianordc 894. Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
(¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
 
Theorempm3.14 748 Theorem *3.14 of [WhiteheadRussell] p. 111. One direction of De Morgan's law). The biconditional holds for decidable propositions as seen at ianordc 894. The converse holds for decidable propositions, as seen at pm3.13dc 954. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑𝜓))
 
Theorempm3.1 749 Theorem *3.1 of [WhiteheadRussell] p. 111. The converse holds for decidable propositions, as seen at anordc 951. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
((𝜑𝜓) → ¬ (¬ 𝜑 ∨ ¬ 𝜓))
 
Theoremjao 750 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)))
 
Theorempm1.2 751 Axiom *1.2 (Taut) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 10-Mar-2013.)
((𝜑𝜑) → 𝜑)
 
Theoremoridm 752 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)
 
Theorempm4.25 753 Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑𝜑))
 
Theoremorim12i 754 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 755 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
(𝜑𝜓)       ((𝜑𝜒) → (𝜓𝜒))
 
Theoremorim2i 756 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
(𝜑𝜓)       ((𝜒𝜑) → (𝜒𝜓))
 
Theoremorbi2i 757 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 758 Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))
 
Theoremorbi12i 759 Infer the disjunction of two equivalences. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))
 
Theorempm1.5 760 Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → (𝜓 ∨ (𝜑𝜒)))
 
Theoremor12 761 Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
((𝜑 ∨ (𝜓𝜒)) ↔ (𝜓 ∨ (𝜑𝜒)))
 
Theoremorass 762 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)))
 
Theorempm2.31 763 Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → ((𝜑𝜓) ∨ 𝜒))
 
Theorempm2.32 764 Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∨ 𝜒) → (𝜑 ∨ (𝜓𝜒)))
 
Theoremor32 765 A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∨ 𝜓))
 
Theoremor4 766 Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∨ (𝜓𝜃)))
 
Theoremor42 767 Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∨ (𝜃𝜓)))
 
Theoremorordi 768 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒)))
 
Theoremorordir 769 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
(((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))
 
Theorempm2.3 770 Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → (𝜑 ∨ (𝜒𝜓)))
 
Theorempm2.41 771 Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((𝜓 ∨ (𝜑𝜓)) → (𝜑𝜓))
 
Theorempm2.42 772 Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 ∨ (𝜑𝜓)) → (𝜑𝜓))
 
Theorempm2.4 773 Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜑𝜓)) → (𝜑𝜓))
 
Theorempm4.44 774 Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑 ∨ (𝜑𝜓)))
 
Theorempm4.56 775 Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
 
Theoremoranim 776 Disjunction in terms of conjunction (DeMorgan's law). One direction of Theorem *4.57 of [WhiteheadRussell] p. 120. The converse does not hold intuitionistically but does hold in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
((𝜑𝜓) → ¬ (¬ 𝜑 ∧ ¬ 𝜓))
 
Theorempm4.78i 777 Implication distributes over disjunction. One direction of Theorem *4.78 of [WhiteheadRussell] p. 121. The converse holds in classical logic. (Contributed by Jim Kingdon, 15-Jan-2018.)
(((𝜑𝜓) ∨ (𝜑𝜒)) → (𝜑 → (𝜓𝜒)))
 
Theoremmtord 778 A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → ¬ 𝜃)    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ¬ 𝜓)
 
Theorempm4.45 779 Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑 ∧ (𝜑𝜓)))
 
Theorempm3.48 780 Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.) (Revised by NM, 1-Dec-2012.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) → (𝜓𝜃)))
 
Theoremorim12d 781 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 10-May-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))
 
Theoremorim1d 782 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) → (𝜒𝜃)))
 
Theoremorim2d 783 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) → (𝜃𝜒)))
 
Theoremorim2 784 Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
((𝜓𝜒) → ((𝜑𝜓) → (𝜑𝜒)))
 
Theoremorbi2d 785 Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) ↔ (𝜃𝜒)))
 
Theoremorbi1d 786 Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜃)))
 
Theoremorbi1 787 Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))
 
Theoremorbi12d 788 Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
 
Theorempm5.61 789 Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.)
(((𝜑𝜓) ∧ ¬ 𝜓) ↔ (𝜑 ∧ ¬ 𝜓))
 
Theoremjaoian 790 Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜃𝜓) → 𝜒)       (((𝜑𝜃) ∧ 𝜓) → 𝜒)
 
Theoremjao1i 791 Add a disjunct in the antecedent of an implication. (Contributed by Rodolfo Medina, 24-Sep-2010.)
(𝜓 → (𝜒𝜑))       ((𝜑𝜓) → (𝜒𝜑))
 
Theoremjaodan 792 Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜒)       ((𝜑 ∧ (𝜓𝜃)) → 𝜒)
 
Theoremmpjaodan 793 Eliminate a disjunction in a deduction. A translation of natural deduction rule E ( elimination). (Contributed by Mario Carneiro, 29-May-2016.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜒)    &   (𝜑 → (𝜓𝜃))       (𝜑𝜒)
 
Theorempm4.77 794 Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
(((𝜓𝜑) ∧ (𝜒𝜑)) ↔ ((𝜓𝜒) → 𝜑))
 
Theorempm2.63 795 Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((¬ 𝜑𝜓) → 𝜓))
 
Theorempm2.64 796 Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑 ∨ ¬ 𝜓) → 𝜑))
 
Theorempm5.53 797 Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((((𝜑𝜓) ∨ 𝜒) → 𝜃) ↔ (((𝜑𝜃) ∧ (𝜓𝜃)) ∧ (𝜒𝜃)))
 
Theorempm2.38 798 Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
((𝜓𝜒) → ((𝜓𝜑) → (𝜒𝜑)))
 
Theorempm2.36 799 Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
((𝜓𝜒) → ((𝜑𝜓) → (𝜒𝜑)))
 
Theorempm2.37 800 Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
((𝜓𝜒) → ((𝜓𝜑) → (𝜑𝜒)))
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