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Theorem List for Intuitionistic Logic Explorer - 701-800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremorc 701 Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.)
 |-  ( ph  ->  ( ph  \/  ps ) )
 
Theorempm2.67-2 702 Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
 |-  ( ( ( ph  \/  ch )  ->  ps )  ->  ( ph  ->  ps )
 )
 
Theoremoibabs 703 Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)
 |-  ( ( ( ph  \/  ps )  ->  ( ph 
 <->  ps ) )  <->  ( ph  <->  ps ) )
 
Theorempm3.44 704 Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
 |-  ( ( ( ps 
 ->  ph )  /\  ( ch  ->  ph ) )  ->  ( ( ps  \/  ch )  ->  ph ) )
 
Theoremjaoi 705 Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) (Revised by NM, 31-Jan-2015.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ps )   =>    |-  ( ( ph  \/  ch )  ->  ps )
 
Theoremjaod 706 Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.) (Revised by NM, 4-Apr-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ch ) )   =>    |-  ( ph  ->  (
 ( ps  \/  th )  ->  ch ) )
 
Theoremmpjaod 707 Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ch ) )   &    |-  ( ph  ->  ( ps  \/  th )
 )   =>    |-  ( ph  ->  ch )
 
Theoremjaao 708 Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ta  ->  ch ) )   =>    |-  ( ( ph  /\  th )  ->  ( ( ps 
 \/  ta )  ->  ch )
 )
 
Theoremjaoa 709 Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ta  ->  ch ) )   =>    |-  ( ( ph  \/  th )  ->  ( ( ps  /\  ta )  ->  ch ) )
 
Theoremimorr 710 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 882. (Contributed by Jim Kingdon, 21-Jul-2018.)
 |-  ( ( -.  ph  \/  ps )  ->  ( ph  ->  ps ) )
 
Theorempm2.53 711 Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 876). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.)
 |-  ( ( ph  \/  ps )  ->  ( -.  ph 
 ->  ps ) )
 
Theoremori 712 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  \/  ps )   =>    |-  ( -.  ph  ->  ps )
 
Theoremord 713 Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ph  ->  ( -.  ps 
 ->  ch ) )
 
Theoremorel1 714 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.)
 |-  ( -.  ph  ->  ( ( ph  \/  ps )  ->  ps ) )
 
Theoremorel2 715 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.)
 |-  ( -.  ph  ->  ( ( ps  \/  ph )  ->  ps ) )
 
Theorempm1.4 716 Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 15-Nov-2012.)
 |-  ( ( ph  \/  ps )  ->  ( ps  \/  ph ) )
 
Theoremorcom 717 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.)
 |-  ( ( ph  \/  ps )  <->  ( ps  \/  ph ) )
 
Theoremorcomd 718 Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
 |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ph  ->  ( ch  \/  ps ) )
 
Theoremorcoms 719 Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
 |-  ( ( ph  \/  ps )  ->  ch )   =>    |-  (
 ( ps  \/  ph )  ->  ch )
 
Theoremorci 720 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ph   =>    |-  ( ph  \/  ps )
 
Theoremolci 721 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ph   =>    |-  ( ps  \/  ph )
 
Theoremorcd 722 Deduction introducing a disjunct. (Contributed by NM, 20-Sep-2007.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ps  \/  ch ) )
 
Theoremolcd 723 Deduction introducing a disjunct. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  ps ) )
 
Theoremorcs 724 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.)
 |-  ( ( ph  \/  ps )  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theoremolcs 725 Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
 |-  ( ( ph  \/  ps )  ->  ch )   =>    |-  ( ps  ->  ch )
 
Theorempm2.07 726 Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  ->  ( ph  \/  ph ) )
 
Theorempm2.45 727 Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  -.  ph )
 
Theorempm2.46 728 Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  -.  ps )
 
Theorempm2.47 729 Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  ( -.  ph  \/  ps )
 )
 
Theorempm2.48 730 Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  ( ph  \/  -.  ps )
 )
 
Theorempm2.49 731 Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  ( -.  ph  \/  -.  ps ) )
 
Theorempm2.67 732 Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
 |-  ( ( ( ph  \/  ps )  ->  ps )  ->  ( ph  ->  ps )
 )
 
Theorembiorf 733 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.)
 |-  ( -.  ph  ->  ( ps  <->  ( ph  \/  ps ) ) )
 
Theorembiortn 734 A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
 |-  ( ph  ->  ( ps 
 <->  ( -.  ph  \/  ps ) ) )
 
Theorembiorfi 735 A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)
 |- 
 -.  ph   =>    |-  ( ps  <->  ( ps  \/  ph ) )
 
Theorempm2.621 736 Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 13-Dec-2013.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  \/  ps )  ->  ps ) )
 
Theorempm2.62 737 Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.)
 |-  ( ( ph  \/  ps )  ->  ( ( ph  ->  ps )  ->  ps )
 )
 
Theoremimorri 738 Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( -.  ph  \/  ps )   =>    |-  ( ph  ->  ps )
 
Theorempm4.52im 739 One direction of theorem *4.52 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.)
 |-  ( ( ph  /\  -.  ps )  ->  -.  ( -.  ph  \/  ps )
 )
 
Theorempm4.53r 740 One direction of theorem *4.53 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.)
 |-  ( ( -.  ph  \/  ps )  ->  -.  ( ph  /\  -.  ps )
 )
 
Theoremioran 741 Negated disjunction in terms of conjunction. This version of DeMorgan's law is a biconditional for all propositions (not just decidable ones), unlike oranim 770, anordc 940, or ianordc 884. Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( -.  ( ph  \/  ps )  <->  ( -.  ph  /\ 
 -.  ps ) )
 
Theorempm3.14 742 Theorem *3.14 of [WhiteheadRussell] p. 111. One direction of De Morgan's law). The biconditional holds for decidable propositions as seen at ianordc 884. The converse holds for decidable propositions, as seen at pm3.13dc 943. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( -.  ph  \/  -.  ps )  ->  -.  ( ph  /\  ps ) )
 
Theorempm3.1 743 Theorem *3.1 of [WhiteheadRussell] p. 111. The converse holds for decidable propositions, as seen at anordc 940. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( ph  /\  ps )  ->  -.  ( -.  ph 
 \/  -.  ps )
 )
 
Theoremjao 744 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.)
 |-  ( ( ph  ->  ps )  ->  ( ( ch  ->  ps )  ->  (
 ( ph  \/  ch )  ->  ps ) ) )
 
Theorempm1.2 745 Axiom *1.2 (Taut) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 10-Mar-2013.)
 |-  ( ( ph  \/  ph )  ->  ph )
 
Theoremoridm 746 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.)
 |-  ( ( ph  \/  ph )  <->  ph )
 
Theorempm4.25 747 Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  <->  ( ph  \/  ph ) )
 
Theoremorim12i 748 Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   =>    |-  ( ( ph  \/  ch )  ->  ( ps  \/  th ) )
 
Theoremorim1i 749 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  \/  ch )  ->  ( ps  \/  ch ) )
 
Theoremorim2i 750 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  \/  ph )  ->  ( ch  \/  ps ) )
 
Theoremorbi2i 751 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.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ch  \/  ph )  <->  ( ch  \/  ps ) )
 
Theoremorbi1i 752 Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ph  \/  ch )  <->  ( ps  \/  ch ) )
 
Theoremorbi12i 753 Infer the disjunction of two equivalences. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  \/  ch )  <->  ( ps  \/  th ) )
 
Theorempm1.5 754 Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 )  ->  ( ps  \/  ( ph  \/  ch ) ) )
 
Theoremor12 755 Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 ) 
 <->  ( ps  \/  ( ph  \/  ch ) ) )
 
Theoremorass 756 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.)
 |-  ( ( ( ph  \/  ps )  \/  ch ) 
 <->  ( ph  \/  ( ps  \/  ch ) ) )
 
Theorempm2.31 757 Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 )  ->  ( ( ph  \/  ps )  \/ 
 ch ) )
 
Theorempm2.32 758 Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  \/  ps )  \/  ch )  ->  ( ph  \/  ( ps  \/  ch )
 ) )
 
Theoremor32 759 A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( ( ph  \/  ps )  \/  ch ) 
 <->  ( ( ph  \/  ch )  \/  ps )
 )
 
Theoremor4 760 Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
 |-  ( ( ( ph  \/  ps )  \/  ( ch  \/  th ) )  <-> 
 ( ( ph  \/  ch )  \/  ( ps 
 \/  th ) ) )
 
Theoremor42 761 Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
 |-  ( ( ( ph  \/  ps )  \/  ( ch  \/  th ) )  <-> 
 ( ( ph  \/  ch )  \/  ( th  \/  ps ) ) )
 
Theoremorordi 762 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 ) 
 <->  ( ( ph  \/  ps )  \/  ( ph  \/  ch ) ) )
 
Theoremorordir 763 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
 |-  ( ( ( ph  \/  ps )  \/  ch ) 
 <->  ( ( ph  \/  ch )  \/  ( ps 
 \/  ch ) ) )
 
Theorempm2.3 764 Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 )  ->  ( ph  \/  ( ch  \/  ps ) ) )
 
Theorempm2.41 765 Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ps  \/  ( ph  \/  ps )
 )  ->  ( ph  \/  ps ) )
 
Theorempm2.42 766 Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( -.  ph  \/  ( ph  ->  ps )
 )  ->  ( ph  ->  ps ) )
 
Theorempm2.4 767 Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ph  \/  ps )
 )  ->  ( ph  \/  ps ) )
 
Theorempm4.44 768 Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  <->  ( ph  \/  ( ph  /\  ps )
 ) )
 
Theorempm4.56 769 Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( -.  ph  /\ 
 -.  ps )  <->  -.  ( ph  \/  ps ) )
 
Theoremoranim 770 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.)
 |-  ( ( ph  \/  ps )  ->  -.  ( -.  ph  /\  -.  ps ) )
 
Theorempm4.78i 771 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.)
 |-  ( ( ( ph  ->  ps )  \/  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  \/  ch ) ) )
 
Theoremmtord 772 A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  -.  th )   &    |-  ( ph  ->  ( ps  ->  ( ch  \/  th ) ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theorempm4.45 773 Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  <->  ( ph  /\  ( ph  \/  ps ) ) )
 
Theorempm3.48 774 Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.) (Revised by NM, 1-Dec-2012.)
 |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  th ) )  ->  ( ( ph  \/  ch )  ->  ( ps  \/  th ) ) )
 
Theoremorim12d 775 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 10-May-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ta ) )   =>    |-  ( ph  ->  (
 ( ps  \/  th )  ->  ( ch  \/  ta ) ) )
 
Theoremorim1d 776 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( ps  \/  th )  ->  ( ch  \/  th ) ) )
 
Theoremorim2d 777 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( th  \/  ps )  ->  ( th  \/  ch ) ) )
 
Theoremorim2 778 Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ps  ->  ch )  ->  ( ( ph  \/  ps )  ->  ( ph  \/  ch )
 ) )
 
Theoremorbi2d 779 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.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( th  \/  ps ) 
 <->  ( th  \/  ch ) ) )
 
Theoremorbi1d 780 Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( ps  \/  th ) 
 <->  ( ch  \/  th ) ) )
 
Theoremorbi1 781 Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ph  \/  ch )  <->  ( ps  \/  ch )
 ) )
 
Theoremorbi12d 782 Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( ps  \/  th ) 
 <->  ( ch  \/  ta ) ) )
 
Theorempm5.61 783 Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.)
 |-  ( ( ( ph  \/  ps )  /\  -.  ps )  <->  ( ph  /\  -.  ps ) )
 
Theoremjaoian 784 Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( th  /\ 
 ps )  ->  ch )   =>    |-  (
 ( ( ph  \/  th )  /\  ps )  ->  ch )
 
Theoremjao1i 785 Add a disjunct in the antecedent of an implication. (Contributed by Rodolfo Medina, 24-Sep-2010.)
 |-  ( ps  ->  ( ch  ->  ph ) )   =>    |-  ( ( ph  \/  ps )  ->  ( ch  ->  ph ) )
 
Theoremjaodan 786 Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 th )  ->  ch )   =>    |-  (
 ( ph  /\  ( ps 
 \/  th ) )  ->  ch )
 
Theoremmpjaodan 787 Eliminate a disjunction in a deduction. A translation of natural deduction rule  \/ E ( \/ elimination). (Contributed by Mario Carneiro, 29-May-2016.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 th )  ->  ch )   &    |-  ( ph  ->  ( ps  \/  th ) )   =>    |-  ( ph  ->  ch )
 
Theorempm4.77 788 Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ps 
 ->  ph )  /\  ( ch  ->  ph ) )  <->  ( ( ps 
 \/  ch )  ->  ph )
 )
 
Theorempm2.63 789 Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ps )  ->  ( ( -.  ph  \/  ps )  ->  ps ) )
 
Theorempm2.64 790 Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ps )  ->  ( ( ph  \/  -.  ps )  -> 
 ph ) )
 
Theorempm5.53 791 Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( (
 ph  \/  ps )  \/  ch )  ->  th )  <->  ( ( ( ph  ->  th )  /\  ( ps 
 ->  th ) )  /\  ( ch  ->  th )
 ) )
 
Theorempm2.38 792 Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
 |-  ( ( ps  ->  ch )  ->  ( ( ps  \/  ph )  ->  ( ch  \/  ph ) ) )
 
Theorempm2.36 793 Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
 |-  ( ( ps  ->  ch )  ->  ( ( ph  \/  ps )  ->  ( ch  \/  ph )
 ) )
 
Theorempm2.37 794 Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
 |-  ( ( ps  ->  ch )  ->  ( ( ps  \/  ph )  ->  ( ph  \/  ch ) ) )
 
Theorempm2.73 795 Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ps )  ->  ( (
 ( ph  \/  ps )  \/  ch )  ->  ( ps  \/  ch ) ) )
 
Theorempm2.74 796 Theorem *2.74 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( ps  ->  ph )  ->  ( (
 ( ph  \/  ps )  \/  ch )  ->  ( ph  \/  ch ) ) )
 
Theorempm2.76 797 Theorem *2.76 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( ph  \/  ( ps  ->  ch )
 )  ->  ( ( ph  \/  ps )  ->  ( ph  \/  ch )
 ) )
 
Theorempm2.75 798 Theorem *2.75 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Jan-2013.)
 |-  ( ( ph  \/  ps )  ->  ( ( ph  \/  ( ps  ->  ch ) )  ->  ( ph  \/  ch ) ) )
 
Theorempm2.8 799 Theorem *2.8 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( ph  \/  ps )  ->  ( ( -.  ps  \/  ch )  ->  ( ph  \/  ch ) ) )
 
Theorempm2.81 800 Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ps  ->  ( ch  ->  th )
 )  ->  ( ( ph  \/  ps )  ->  ( ( ph  \/  ch )  ->  ( ph  \/  th ) ) ) )
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