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Theorem List for Intuitionistic Logic Explorer - 701-800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnbn3 701 Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
 |-  ph   =>    |-  ( -.  ps  <->  ( ps  <->  -.  ph ) )
 
Theorem2false 702 Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |- 
 -.  ph   &    |-  -.  ps   =>    |-  ( ph  <->  ps )
 
Theorem2falsed 703 Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ph  ->  -.  ch )   =>    |-  ( ph  ->  ( ps 
 <->  ch ) )
 
Theorempm5.21ni 704 Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ps )   =>    |-  ( -.  ps  ->  (
 ph 
 <->  ch ) )
 
Theorempm5.21nii 705 Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ps )   &    |-  ( ps  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ph  <->  ch )
 
Theorempm5.21ndd 706 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.)
 |-  ( ph  ->  ( ch  ->  ps ) )   &    |-  ( ph  ->  ( th  ->  ps ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )   =>    |-  ( ph  ->  ( ch  <->  th ) )
 
Theorempm5.19 707 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |- 
 -.  ( ph  <->  -.  ph )
 
Theorempm4.8 708 Theorem *4.8 of [WhiteheadRussell] p. 122. This one holds for all propositions, but compare with pm4.81dc 909 which requires a decidability condition. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  -.  ph )  <->  -.  ph )
 
1.2.6  Logical disjunction
 
Syntaxwo 709 Extend wff definition to include disjunction ('or').
 wff  ( ph  \/  ps )
 
Axiomax-io 710 Definition of 'or'. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias jaob 711 instead. (New usage is discouraged.)
 |-  ( ( ( ph  \/  ch )  ->  ps )  <->  ( ( ph  ->  ps )  /\  ( ch  ->  ps )
 ) )
 
Theoremjaob 711 Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. Alias of ax-io 710. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( ( ph  \/  ch )  ->  ps )  <->  ( ( ph  ->  ps )  /\  ( ch  ->  ps )
 ) )
 
Theoremolc 712 Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.)
 |-  ( ph  ->  ( ps  \/  ph ) )
 
Theoremorc 713 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 ) )
 
Theorempm2.67-2 714 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 715 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 ) )
 
Theorempm3.44 716 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 717 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 718 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 719 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 720 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 721 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 722 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 898. (Contributed by Jim Kingdon, 21-Jul-2018.)
 |-  ( ( -.  ph  \/  ps )  ->  ( ph  ->  ps ) )
 
Theorempm2.53 723 Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 892). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.)
 |-  ( ( ph  \/  ps )  ->  ( -.  ph 
 ->  ps ) )
 
Theoremori 724 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  \/  ps )   =>    |-  ( -.  ph  ->  ps )
 
Theoremord 725 Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ph  ->  ( -.  ps 
 ->  ch ) )
 
Theoremorel1 726 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 727 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 ) )
 
Theorempm1.4 728 Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 15-Nov-2012.)
 |-  ( ( ph  \/  ps )  ->  ( ps  \/  ph ) )
 
Theoremorcom 729 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 730 Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
 |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ph  ->  ( ch  \/  ps ) )
 
Theoremorcoms 731 Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
 |-  ( ( ph  \/  ps )  ->  ch )   =>    |-  (
 ( ps  \/  ph )  ->  ch )
 
Theoremorci 732 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ph   =>    |-  ( ph  \/  ps )
 
Theoremolci 733 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ph   =>    |-  ( ps  \/  ph )
 
Theoremorcd 734 Deduction introducing a disjunct. (Contributed by NM, 20-Sep-2007.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ps  \/  ch ) )
 
Theoremolcd 735 Deduction introducing a disjunct. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  ps ) )
 
Theoremorcs 736 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 737 Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
 |-  ( ( ph  \/  ps )  ->  ch )   =>    |-  ( ps  ->  ch )
 
Theorempm2.07 738 Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  ->  ( ph  \/  ph ) )
 
Theorempm2.45 739 Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  -.  ph )
 
Theorempm2.46 740 Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  -.  ps )
 
Theorempm2.47 741 Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  ( -.  ph  \/  ps )
 )
 
Theorempm2.48 742 Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  ( ph  \/  -.  ps )
 )
 
Theorempm2.49 743 Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  ( -.  ph  \/  -.  ps ) )
 
Theorempm2.67 744 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 745 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 746 A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
 |-  ( ph  ->  ( ps 
 <->  ( -.  ph  \/  ps ) ) )
 
Theorembiorfi 747 A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)
 |- 
 -.  ph   =>    |-  ( ps  <->  ( ps  \/  ph ) )
 
Theorempm2.621 748 Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 13-Dec-2013.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  \/  ps )  ->  ps ) )
 
Theorempm2.62 749 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 750 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 751 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 752 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 753 Negated disjunction in terms of conjunction. This version of DeMorgan's law is a biconditional for all propositions (not just decidable ones), unlike oranim 782, anordc 957, or ianordc 900. 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 ) )
 
Theorempm3.14 754 Theorem *3.14 of [WhiteheadRussell] p. 111. One direction of De Morgan's law). The biconditional holds for decidable propositions as seen at ianordc 900. The converse holds for decidable propositions, as seen at pm3.13dc 960. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( -.  ph  \/  -.  ps )  ->  -.  ( ph  /\  ps ) )
 
Theorempm3.1 755 Theorem *3.1 of [WhiteheadRussell] p. 111. The converse holds for decidable propositions, as seen at anordc 957. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( ph  /\  ps )  ->  -.  ( -.  ph 
 \/  -.  ps )
 )
 
Theoremjao 756 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 ) ) )
 
Theorempm1.2 757 Axiom *1.2 (Taut) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 10-Mar-2013.)
 |-  ( ( ph  \/  ph )  ->  ph )
 
Theoremoridm 758 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 )
 
Theorempm4.25 759 Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  <->  ( ph  \/  ph ) )
 
Theoremorim12i 760 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 761 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  \/  ch )  ->  ( ps  \/  ch ) )
 
Theoremorim2i 762 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  \/  ph )  ->  ( ch  \/  ps ) )
 
Theoremorbi2i 763 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 764 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 765 Infer the disjunction of two equivalences. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  \/  ch )  <->  ( ps  \/  th ) )
 
Theorempm1.5 766 Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 )  ->  ( ps  \/  ( ph  \/  ch ) ) )
 
Theoremor12 767 Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 ) 
 <->  ( ps  \/  ( ph  \/  ch ) ) )
 
Theoremorass 768 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 ) ) )
 
Theorempm2.31 769 Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 )  ->  ( ( ph  \/  ps )  \/ 
 ch ) )
 
Theorempm2.32 770 Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  \/  ps )  \/  ch )  ->  ( ph  \/  ( ps  \/  ch )
 ) )
 
Theoremor32 771 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 772 Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
 |-  ( ( ( ph  \/  ps )  \/  ( ch  \/  th ) )  <-> 
 ( ( ph  \/  ch )  \/  ( ps 
 \/  th ) ) )
 
Theoremor42 773 Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
 |-  ( ( ( ph  \/  ps )  \/  ( ch  \/  th ) )  <-> 
 ( ( ph  \/  ch )  \/  ( th  \/  ps ) ) )
 
Theoremorordi 774 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 ) 
 <->  ( ( ph  \/  ps )  \/  ( ph  \/  ch ) ) )
 
Theoremorordir 775 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
 |-  ( ( ( ph  \/  ps )  \/  ch ) 
 <->  ( ( ph  \/  ch )  \/  ( ps 
 \/  ch ) ) )
 
Theorempm2.3 776 Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 )  ->  ( ph  \/  ( ch  \/  ps ) ) )
 
Theorempm2.41 777 Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ps  \/  ( ph  \/  ps )
 )  ->  ( ph  \/  ps ) )
 
Theorempm2.42 778 Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( -.  ph  \/  ( ph  ->  ps )
 )  ->  ( ph  ->  ps ) )
 
Theorempm2.4 779 Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ph  \/  ps )
 )  ->  ( ph  \/  ps ) )
 
Theorempm4.44 780 Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  <->  ( ph  \/  ( ph  /\  ps )
 ) )
 
Theorempm4.56 781 Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( -.  ph  /\ 
 -.  ps )  <->  -.  ( ph  \/  ps ) )
 
Theoremoranim 782 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 783 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 784 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 )
 
Theorempm4.45 785 Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  <->  ( ph  /\  ( ph  \/  ps ) ) )
 
Theorempm3.48 786 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 787 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 788 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( ps  \/  th )  ->  ( ch  \/  th ) ) )
 
Theoremorim2d 789 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( th  \/  ps )  ->  ( th  \/  ch ) ) )
 
Theoremorim2 790 Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ps  ->  ch )  ->  ( ( ph  \/  ps )  ->  ( ph  \/  ch )
 ) )
 
Theoremorbi2d 791 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 792 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 793 Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ph  \/  ch )  <->  ( ps  \/  ch )
 ) )
 
Theoremorbi12d 794 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 ) ) )
 
Theorempm5.61 795 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 796 Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( th  /\ 
 ps )  ->  ch )   =>    |-  (
 ( ( ph  \/  th )  /\  ps )  ->  ch )
 
Theoremjao1i 797 Add a disjunct in the antecedent of an implication. (Contributed by Rodolfo Medina, 24-Sep-2010.)
 |-  ( ps  ->  ( ch  ->  ph ) )   =>    |-  ( ( ph  \/  ps )  ->  ( ch  ->  ph ) )
 
Theoremjaodan 798 Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 th )  ->  ch )   =>    |-  (
 ( ph  /\  ( ps 
 \/  th ) )  ->  ch )
 
Theoremmpjaodan 799 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 )
 
Theorempm4.77 800 Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ps 
 ->  ph )  /\  ( ch  ->  ph ) )  <->  ( ( ps 
 \/  ch )  ->  ph )
 )
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