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Theorem List for Intuitionistic Logic Explorer - 701-800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempm1.4 701 Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 15-Nov-2012.)
 |-  ( ( ph  \/  ps )  ->  ( ps  \/  ph ) )
 
Theoremorcom 702 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 703 Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
 |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ph  ->  ( ch  \/  ps ) )
 
Theoremorcoms 704 Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
 |-  ( ( ph  \/  ps )  ->  ch )   =>    |-  (
 ( ps  \/  ph )  ->  ch )
 
Theoremorci 705 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ph   =>    |-  ( ph  \/  ps )
 
Theoremolci 706 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ph   =>    |-  ( ps  \/  ph )
 
Theoremorcd 707 Deduction introducing a disjunct. (Contributed by NM, 20-Sep-2007.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ps  \/  ch ) )
 
Theoremolcd 708 Deduction introducing a disjunct. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  ps ) )
 
Theoremorcs 709 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 710 Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
 |-  ( ( ph  \/  ps )  ->  ch )   =>    |-  ( ps  ->  ch )
 
Theorempm2.07 711 Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  ->  ( ph  \/  ph ) )
 
Theorempm2.45 712 Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  -.  ph )
 
Theorempm2.46 713 Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  -.  ps )
 
Theorempm2.47 714 Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  ( -.  ph  \/  ps )
 )
 
Theorempm2.48 715 Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  ( ph  \/  -.  ps )
 )
 
Theorempm2.49 716 Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  ( -.  ph  \/  -.  ps ) )
 
Theorempm2.67 717 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 718 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 719 A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
 |-  ( ph  ->  ( ps 
 <->  ( -.  ph  \/  ps ) ) )
 
Theorembiorfi 720 A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)
 |- 
 -.  ph   =>    |-  ( ps  <->  ( ps  \/  ph ) )
 
Theorempm2.621 721 Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 13-Dec-2013.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  \/  ps )  ->  ps ) )
 
Theorempm2.62 722 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 723 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 724 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 725 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 726 Negated disjunction in terms of conjunction. This version of DeMorgan's law is a biconditional for all propositions (not just decidable ones), unlike oranim 755, anordc 925, or ianordc 869. 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 ) )
 
Theorempm3.14 727 Theorem *3.14 of [WhiteheadRussell] p. 111. One direction of De Morgan's law). The biconditional holds for decidable propositions as seen at ianordc 869. The converse holds for decidable propositions, as seen at pm3.13dc 928. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( -.  ph  \/  -.  ps )  ->  -.  ( ph  /\  ps ) )
 
Theorempm3.1 728 Theorem *3.1 of [WhiteheadRussell] p. 111. The converse holds for decidable propositions, as seen at anordc 925. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( ph  /\  ps )  ->  -.  ( -.  ph 
 \/  -.  ps )
 )
 
Theoremjao 729 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 ) ) )
 
Theorempm1.2 730 Axiom *1.2 (Taut) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 10-Mar-2013.)
 |-  ( ( ph  \/  ph )  ->  ph )
 
Theoremoridm 731 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 )
 
Theorempm4.25 732 Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  <->  ( ph  \/  ph ) )
 
Theoremorim12i 733 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 734 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  \/  ch )  ->  ( ps  \/  ch ) )
 
Theoremorim2i 735 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  \/  ph )  ->  ( ch  \/  ps ) )
 
Theoremorbi2i 736 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 737 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 738 Infer the disjunction of two equivalences. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  \/  ch )  <->  ( ps  \/  th ) )
 
Theorempm1.5 739 Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 )  ->  ( ps  \/  ( ph  \/  ch ) ) )
 
Theoremor12 740 Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 ) 
 <->  ( ps  \/  ( ph  \/  ch ) ) )
 
Theoremorass 741 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 ) ) )
 
Theorempm2.31 742 Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 )  ->  ( ( ph  \/  ps )  \/ 
 ch ) )
 
Theorempm2.32 743 Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  \/  ps )  \/  ch )  ->  ( ph  \/  ( ps  \/  ch )
 ) )
 
Theoremor32 744 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 745 Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
 |-  ( ( ( ph  \/  ps )  \/  ( ch  \/  th ) )  <-> 
 ( ( ph  \/  ch )  \/  ( ps 
 \/  th ) ) )
 
Theoremor42 746 Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
 |-  ( ( ( ph  \/  ps )  \/  ( ch  \/  th ) )  <-> 
 ( ( ph  \/  ch )  \/  ( th  \/  ps ) ) )
 
Theoremorordi 747 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 ) 
 <->  ( ( ph  \/  ps )  \/  ( ph  \/  ch ) ) )
 
Theoremorordir 748 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
 |-  ( ( ( ph  \/  ps )  \/  ch ) 
 <->  ( ( ph  \/  ch )  \/  ( ps 
 \/  ch ) ) )
 
Theorempm2.3 749 Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 )  ->  ( ph  \/  ( ch  \/  ps ) ) )
 
Theorempm2.41 750 Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ps  \/  ( ph  \/  ps )
 )  ->  ( ph  \/  ps ) )
 
Theorempm2.42 751 Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( -.  ph  \/  ( ph  ->  ps )
 )  ->  ( ph  ->  ps ) )
 
Theorempm2.4 752 Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ph  \/  ps )
 )  ->  ( ph  \/  ps ) )
 
Theorempm4.44 753 Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  <->  ( ph  \/  ( ph  /\  ps )
 ) )
 
Theorempm4.56 754 Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( -.  ph  /\ 
 -.  ps )  <->  -.  ( ph  \/  ps ) )
 
Theoremoranim 755 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 756 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 757 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 )
 
Theorempm4.45 758 Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  <->  ( ph  /\  ( ph  \/  ps ) ) )
 
Theorempm3.48 759 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 760 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 761 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( ps  \/  th )  ->  ( ch  \/  th ) ) )
 
Theoremorim2d 762 Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( th  \/  ps )  ->  ( th  \/  ch ) ) )
 
Theoremorim2 763 Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ps  ->  ch )  ->  ( ( ph  \/  ps )  ->  ( ph  \/  ch )
 ) )
 
Theoremorbi2d 764 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 765 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 766 Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ph  \/  ch )  <->  ( ps  \/  ch )
 ) )
 
Theoremorbi12d 767 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 ) ) )
 
Theorempm5.61 768 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 769 Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( th  /\ 
 ps )  ->  ch )   =>    |-  (
 ( ( ph  \/  th )  /\  ps )  ->  ch )
 
Theoremjao1i 770 Add a disjunct in the antecedent of an implication. (Contributed by Rodolfo Medina, 24-Sep-2010.)
 |-  ( ps  ->  ( ch  ->  ph ) )   =>    |-  ( ( ph  \/  ps )  ->  ( ch  ->  ph ) )
 
Theoremjaodan 771 Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 th )  ->  ch )   =>    |-  (
 ( ph  /\  ( ps 
 \/  th ) )  ->  ch )
 
Theoremmpjaodan 772 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 )
 
Theorempm4.77 773 Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ps 
 ->  ph )  /\  ( ch  ->  ph ) )  <->  ( ( ps 
 \/  ch )  ->  ph )
 )
 
Theorempm2.63 774 Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ps )  ->  ( ( -.  ph  \/  ps )  ->  ps ) )
 
Theorempm2.64 775 Theorem *2.64 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ps )  ->  ( ( ph  \/  -.  ps )  -> 
 ph ) )
 
Theorempm5.53 776 Theorem *5.53 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( (
 ph  \/  ps )  \/  ch )  ->  th )  <->  ( ( ( ph  ->  th )  /\  ( ps 
 ->  th ) )  /\  ( ch  ->  th )
 ) )
 
Theorempm2.38 777 Theorem *2.38 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
 |-  ( ( ps  ->  ch )  ->  ( ( ps  \/  ph )  ->  ( ch  \/  ph ) ) )
 
Theorempm2.36 778 Theorem *2.36 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
 |-  ( ( ps  ->  ch )  ->  ( ( ph  \/  ps )  ->  ( ch  \/  ph )
 ) )
 
Theorempm2.37 779 Theorem *2.37 of [WhiteheadRussell] p. 105. (Contributed by NM, 6-Mar-2008.)
 |-  ( ( ps  ->  ch )  ->  ( ( ps  \/  ph )  ->  ( ph  \/  ch ) ) )
 
Theorempm2.73 780 Theorem *2.73 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ps )  ->  ( (
 ( ph  \/  ps )  \/  ch )  ->  ( ps  \/  ch ) ) )
 
Theorempm2.74 781 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 ) ) )
 
Theorempm2.76 782 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 )
 ) )
 
Theorempm2.75 783 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 ) ) )
 
Theorempm2.8 784 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 ) ) )
 
Theorempm2.81 785 Theorem *2.81 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ps  ->  ( ch  ->  th )
 )  ->  ( ( ph  \/  ps )  ->  ( ( ph  \/  ch )  ->  ( ph  \/  th ) ) ) )
 
Theorempm2.82 786 Theorem *2.82 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  \/  ps )  \/  ch )  ->  ( ( (
 ph  \/  -.  ch )  \/  th )  ->  (
 ( ph  \/  ps )  \/  th ) ) )
 
Theorempm3.2ni 787 Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.)
 |- 
 -.  ph   &    |-  -.  ps   =>    |-  -.  ( ph  \/  ps )
 
Theoremorabs 788 Absorption of redundant internal disjunct. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 28-Feb-2014.)
 |-  ( ph  <->  ( ( ph  \/  ps )  /\  ph )
 )
 
Theoremoranabs 789 Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton, 23-Jun-2005.) (Proof shortened by Wolf Lammen, 10-Nov-2013.)
 |-  ( ( ( ph  \/  -.  ps )  /\  ps )  <->  ( ph  /\  ps ) )
 
Theoremordi 790 Distributive law for disjunction. Theorem *4.41 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( ph  \/  ( ps  /\  ch )
 ) 
 <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch ) ) )
 
Theoremordir 791 Distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
 |-  ( ( ( ph  /\ 
 ps )  \/  ch ) 
 <->  ( ( ph  \/  ch )  /\  ( ps 
 \/  ch ) ) )
 
Theoremandi 792 Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
 |-  ( ( ph  /\  ( ps  \/  ch ) )  <-> 
 ( ( ph  /\  ps )  \/  ( ph  /\  ch ) ) )
 
Theoremandir 793 Distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
 |-  ( ( ( ph  \/  ps )  /\  ch ) 
 <->  ( ( ph  /\  ch )  \/  ( ps  /\  ch ) ) )
 
Theoremorddi 794 Double distributive law for disjunction. (Contributed by NM, 12-Aug-1994.)
 |-  ( ( ( ph  /\ 
 ps )  \/  ( ch  /\  th ) )  <-> 
 ( ( ( ph  \/  ch )  /\  ( ph  \/  th ) ) 
 /\  ( ( ps 
 \/  ch )  /\  ( ps  \/  th ) ) ) )
 
Theoremanddi 795 Double distributive law for conjunction. (Contributed by NM, 12-Aug-1994.)
 |-  ( ( ( ph  \/  ps )  /\  ( ch  \/  th ) )  <-> 
 ( ( ( ph  /\ 
 ch )  \/  ( ph  /\  th ) )  \/  ( ( ps 
 /\  ch )  \/  ( ps  /\  th ) ) ) )
 
Theorempm4.39 796 Theorem *4.39 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  <->  ch )  /\  ( ps  <->  th ) )  ->  ( ( ph  \/  ps )  <->  ( ch  \/  th ) ) )
 
Theorempm4.72 797 Implication in terms of biconditional and disjunction. Theorem *4.72 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Jan-2013.)
 |-  ( ( ph  ->  ps )  <->  ( ps  <->  ( ph  \/  ps ) ) )
 
Theorempm5.16 798 Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |- 
 -.  ( ( ph  <->  ps )  /\  ( ph  <->  -.  ps ) )
 
Theorembiort 799 A wff is disjoined with truth is true. (Contributed by NM, 23-May-1999.)
 |-  ( ph  ->  ( ph 
 <->  ( ph  \/  ps ) ) )
 
1.2.7  Stable propositions
 
Syntaxwstab 800 Extend wff definition to include stability.
 wff STAB  ph
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