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Theorem List for Intuitionistic Logic Explorer - 601-700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempm2.65i 601 Inference rule for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  -.  ps )   =>    |-  -.  ph
 
Theoremmt2 602 A rule similar to modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Sep-2013.)
 |- 
 ps   &    |-  ( ph  ->  -.  ps )   =>    |- 
 -.  ph
 
Theorembiijust 603 Theorem used to justify definition of intuitionistic biconditional df-bi 115. (Contributed by NM, 24-Nov-2017.)
 |-  ( ( ( (
 ph  ->  ps )  /\  ( ps  ->  ph ) )  ->  ( ( ph  ->  ps )  /\  ( ps 
 ->  ph ) ) ) 
 /\  ( ( (
 ph  ->  ps )  /\  ( ps  ->  ph ) )  ->  ( ( ph  ->  ps )  /\  ( ps 
 ->  ph ) ) ) )
 
Theoremcon3 604 Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Feb-2013.)
 |-  ( ( ph  ->  ps )  ->  ( -.  ps 
 ->  -.  ph ) )
 
Theoremcon2 605 Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2013.)
 |-  ( ( ph  ->  -. 
 ps )  ->  ( ps  ->  -.  ph ) )
 
Theoremmt2i 606 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
 |- 
 ch   &    |-  ( ph  ->  ( ps  ->  -.  ch )
 )   =>    |-  ( ph  ->  -.  ps )
 
Theoremnotnoti 607 Infer double negation. (Contributed by NM, 27-Feb-2008.)
 |-  ph   =>    |- 
 -.  -.  ph
 
Theorempm2.21i 608 A contradiction implies anything. Inference from pm2.21 580. (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |- 
 -.  ph   =>    |-  ( ph  ->  ps )
 
Theorempm2.24ii 609 A contradiction implies anything. Inference from pm2.24 584. (Contributed by NM, 27-Feb-2008.)
 |-  ph   &    |- 
 -.  ph   =>    |- 
 ps
 
Theoremnsyld 610 A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
 |-  ( ph  ->  ( ps  ->  -.  ch )
 )   &    |-  ( ph  ->  ( ta  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  -.  ta )
 )
 
Theoremnsyli 611 A negated syllogism inference. (Contributed by NM, 3-May-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  -.  ch )   =>    |-  ( ph  ->  ( th  ->  -. 
 ps ) )
 
Theoremmth8 612 Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
 |-  ( ph  ->  ( -.  ps  ->  -.  ( ph  ->  ps ) ) )
 
Theoremjc 613 Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  -.  ( ps  ->  -.  ch )
 )
 
Theorempm2.51 614 Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  ->  ps )  ->  ( ph  ->  -.  ps )
 )
 
Theorempm2.52 615 Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( -.  ( ph  ->  ps )  ->  ( -.  ph  ->  -.  ps )
 )
 
Theoremexpt 616 Exportation theorem expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( -.  ( ph  ->  -.  ps )  ->  ch )  ->  ( ph  ->  ( ps  ->  ch ) ) )
 
Theoremjarl 617 Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.)
 |-  ( ( ( ph  ->  ps )  ->  ch )  ->  ( -.  ph  ->  ch ) )
 
Theorempm2.65 618 Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. Proofs, such as this one, which assume a proposition, here  ph, derive a contradiction, and therefore conclude  -. 
ph, are valid intuitionistically (and can be called "proof of negation", for example by Section 1.2 of [Bauer] p. 482). By contrast, proofs which assume  -.  ph, derive a contradiction, and conclude  ph, such as condandc 811, are only valid for decidable propositions. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  ->  -.  ps )  ->  -.  ph ) )
 
Theorempm2.65d 619 Deduction rule for proof by contradiction. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 26-May-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  -. 
 ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theorempm2.65da 620 Deduction rule for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 ps )  ->  -.  ch )   =>    |-  ( ph  ->  -.  ps )
 
Theoremmto 621 The rule of modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
 |- 
 -.  ps   &    |-  ( ph  ->  ps )   =>    |- 
 -.  ph
 
Theoremmtod 622 Modus tollens deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremmtoi 623 Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
 |- 
 -.  ch   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  -.  ps )
 
Theoremmtand 624 A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ph  ->  -.  ps )
 
Theoremnotbid 625 Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( -.  ps  <->  -.  ch ) )
 
Theoremcon2b 626 Contraposition. Bidirectional version of con2 605. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  -. 
 ps )  <->  ( ps  ->  -.  ph ) )
 
Theoremnotbii 627 Negate both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  <->  ps )   =>    |-  ( -.  ph  <->  -.  ps )
 
Theoremmtbi 628 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
 |- 
 -.  ph   &    |-  ( ph  <->  ps )   =>    |- 
 -.  ps
 
Theoremmtbir 629 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 14-Oct-2012.)
 |- 
 -.  ps   &    |-  ( ph  <->  ps )   =>    |- 
 -.  ph
 
Theoremmtbid 630 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  -. 
 ch )
 
Theoremmtbird 631 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremmtbii 632 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
 |- 
 -.  ps   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  -. 
 ch )
 
Theoremmtbiri 633 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
 |- 
 -.  ch   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremsylnib 634 A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ps  <->  ch )   =>    |-  ( ph  ->  -.  ch )
 
Theoremsylnibr 635 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.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ch  <->  ps )   =>    |-  ( ph  ->  -.  ch )
 
Theoremsylnbi 636 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.)
 |-  ( ph  <->  ps )   &    |-  ( -.  ps  ->  ch )   =>    |-  ( -.  ph  ->  ch )
 
Theoremsylnbir 637 A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
 |-  ( ps  <->  ph )   &    |-  ( -.  ps  ->  ch )   =>    |-  ( -.  ph  ->  ch )
 
Theoremxchnxbi 638 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( -.  ph  <->  ps )   &    |-  ( ph  <->  ch )   =>    |-  ( -.  ch  <->  ps )
 
Theoremxchnxbir 639 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( -.  ph  <->  ps )   &    |-  ( ch  <->  ph )   =>    |-  ( -.  ch  <->  ps )
 
Theoremxchbinx 640 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( ph  <->  -.  ps )   &    |-  ( ps 
 <->  ch )   =>    |-  ( ph  <->  -.  ch )
 
Theoremxchbinxr 641 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( ph  <->  -.  ps )   &    |-  ( ch 
 <->  ps )   =>    |-  ( ph  <->  -.  ch )
 
Theoremmt2bi 642 A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)
 |-  ph   =>    |-  ( -.  ps  <->  ( ps  ->  -.  ph ) )
 
Theoremmtt 643 Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( -.  ph  ->  ( -.  ps  <->  ( ps  ->  ph ) ) )
 
Theorempm5.21 644 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.)
 |-  ( ( -.  ph  /\ 
 -.  ps )  ->  ( ph 
 <->  ps ) )
 
Theorempm5.21im 645 Two propositions are equivalent if they are both false. Closed form of 2false 650. Equivalent to a bi2 128-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( -.  ph  ->  ( -.  ps  ->  ( ph 
 <->  ps ) ) )
 
Theoremnbn2 646 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.)
 |-  ( -.  ph  ->  ( -.  ps  <->  ( ph  <->  ps ) ) )
 
Theorembibif 647 Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
 |-  ( -.  ps  ->  ( ( ph  <->  ps )  <->  -.  ph ) )
 
Theoremnbn 648 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.)
 |- 
 -.  ph   =>    |-  ( -.  ps  <->  ( ps  <->  ph ) )
 
Theoremnbn3 649 Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
 |-  ph   =>    |-  ( -.  ps  <->  ( ps  <->  -.  ph ) )
 
Theorem2false 650 Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |- 
 -.  ph   &    |-  -.  ps   =>    |-  ( ph  <->  ps )
 
Theorem2falsed 651 Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ph  ->  -.  ch )   =>    |-  ( ph  ->  ( ps 
 <->  ch ) )
 
Theorempm5.21ni 652 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 653 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 654 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 ) )
 
Theorempm5.19 655 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |- 
 -.  ( ph  <->  -.  ph )
 
Theorempm4.8 656 Theorem *4.8 of [WhiteheadRussell] p. 122. This one holds for all propositions, but compare with pm4.81dc 850 which requires a decidability condition. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  -.  ph )  <->  -.  ph )
 
Theoremimnan 657 Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.)
 |-  ( ( ph  ->  -. 
 ps )  <->  -.  ( ph  /\  ps ) )
 
Theoremimnani 658 Express implication in terms of conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
 |- 
 -.  ( ph  /\  ps )   =>    |-  ( ph  ->  -.  ps )
 
Theoremnan 659 Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
 |-  ( ( ph  ->  -.  ( ps  /\  ch ) )  <->  ( ( ph  /\ 
 ps )  ->  -.  ch ) )
 
Theorempm3.24 660 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.)
 |- 
 -.  ( ph  /\  -.  ph )
 
Theoremnotnotnot 661 Triple negation. (Contributed by Jim Kingdon, 28-Jul-2018.)
 |-  ( -.  -.  -.  ph  <->  -.  ph )
 
1.2.6  Logical disjunction
 
Syntaxwo 662 Extend wff definition to include disjunction ('or').
 wff  ( ph  \/  ps )
 
Axiomax-io 663 Definition of 'or'. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias jaob 664 instead. (New usage is discouraged.)
 |-  ( ( ( ph  \/  ch )  ->  ps )  <->  ( ( ph  ->  ps )  /\  ( ch  ->  ps )
 ) )
 
Theoremjaob 664 Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. Alias of ax-io 663. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( ( ph  \/  ch )  ->  ps )  <->  ( ( ph  ->  ps )  /\  ( ch  ->  ps )
 ) )
 
Theoremolc 665 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 666 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 ) )
 
Theorempm2.67-2 667 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 )
 )
 
Theorempm3.44 668 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 669 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 670 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 671 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 672 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 673 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 ) )
 
Theorempm2.53 674 Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 826). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.)
 |-  ( ( ph  \/  ps )  ->  ( -.  ph 
 ->  ps ) )
 
Theoremori 675 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  \/  ps )   =>    |-  ( -.  ph  ->  ps )
 
Theoremord 676 Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ph  ->  ( -.  ps 
 ->  ch ) )
 
Theoremorel1 677 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 678 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 ) )
 
Theorempm1.4 679 Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 15-Nov-2012.)
 |-  ( ( ph  \/  ps )  ->  ( ps  \/  ph ) )
 
Theoremorcom 680 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 681 Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
 |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ph  ->  ( ch  \/  ps ) )
 
Theoremorcoms 682 Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
 |-  ( ( ph  \/  ps )  ->  ch )   =>    |-  (
 ( ps  \/  ph )  ->  ch )
 
Theoremorci 683 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ph   =>    |-  ( ph  \/  ps )
 
Theoremolci 684 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ph   =>    |-  ( ps  \/  ph )
 
Theoremorcd 685 Deduction introducing a disjunct. (Contributed by NM, 20-Sep-2007.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ps  \/  ch ) )
 
Theoremolcd 686 Deduction introducing a disjunct. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  ps ) )
 
Theoremorcs 687 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 688 Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
 |-  ( ( ph  \/  ps )  ->  ch )   =>    |-  ( ps  ->  ch )
 
Theorempm2.07 689 Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  ->  ( ph  \/  ph ) )
 
Theorempm2.45 690 Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  -.  ph )
 
Theorempm2.46 691 Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  -.  ps )
 
Theorempm2.47 692 Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  ( -.  ph  \/  ps )
 )
 
Theorempm2.48 693 Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  ( ph  \/  -.  ps )
 )
 
Theorempm2.49 694 Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  \/  ps )  ->  ( -.  ph  \/  -.  ps ) )
 
Theorempm2.67 695 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 696 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 697 A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
 |-  ( ph  ->  ( ps 
 <->  ( -.  ph  \/  ps ) ) )
 
Theorembiorfi 698 A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)
 |- 
 -.  ph   =>    |-  ( ps  <->  ( ps  \/  ph ) )
 
Theorempm2.621 699 Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 13-Dec-2013.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  \/  ps )  ->  ps ) )
 
Theorempm2.62 700 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 )
 )
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