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Theorem List for Intuitionistic Logic Explorer - 601-700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcon2i 601 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 13-Jun-2013.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ps  ->  -.  ph )
 
Theoremnsyl 602 A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ch  ->  ps )   =>    |-  ( ph  ->  -.  ch )
 
Theoremnotnot 603 Double negation introduction. Theorem *2.12 of [WhiteheadRussell] p. 101. The converse need not hold. It holds exactly for stable propositions (by definition, see df-stab 801) and in particular for decidable propositions (see notnotrdc 813). See also notnotnot 608. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
 |-  ( ph  ->  -.  -.  ph )
 
Theoremnotnotd 604 Deduction associated with notnot 603 and notnoti 619. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  -.  -.  ps )
 
Theoremcon3d 605 A contraposition deduction. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( -.  ch  ->  -.  ps ) )
 
Theoremcon3i 606 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
 |-  ( ph  ->  ps )   =>    |-  ( -.  ps  ->  -.  ph )
 
Theoremcon3rr3 607 Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( -.  ch  ->  ( ph  ->  -.  ps ) )
 
Theoremnotnotnot 608 Triple negation is equivalent to negation. (Contributed by Jim Kingdon, 28-Jul-2018.)
 |-  ( -.  -.  -.  ph  <->  -.  ph )
 
Theoremcon3dimp 609 Variant of con3d 605 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ph  /\ 
 -.  ch )  ->  -.  ps )
 
Theorempm2.01da 610 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ph  /\  ps )  ->  -.  ps )   =>    |-  ( ph  ->  -.  ps )
 
Theorempm3.2im 611 In classical logic, this is just a restatement of pm3.2 138. In intuitionistic logic, it still holds, but is weaker than pm3.2. (Contributed by Mario Carneiro, 12-May-2015.)
 |-  ( ph  ->  ( ps  ->  -.  ( ph  ->  -.  ps ) ) )
 
Theoremexpi 612 An exportation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)
 |-  ( -.  ( ph  ->  -.  ps )  ->  ch )   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theorempm2.65i 613 Inference for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  -.  ps )   =>    |-  -.  ph
 
Theoremmt2 614 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 615 Theorem used to justify definition of intuitionistic biconditional df-bi 116. (Contributed by NM, 24-Nov-2017.)
 |-  ( ( ( (
 ph  ->  ps )  /\  ( ps  ->  ph ) )  ->  ( ( ph  ->  ps )  /\  ( ps 
 ->  ph ) ) ) 
 /\  ( ( (
 ph  ->  ps )  /\  ( ps  ->  ph ) )  ->  ( ( ph  ->  ps )  /\  ( ps 
 ->  ph ) ) ) )
 
Theoremcon3 616 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 617 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 618 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
 |- 
 ch   &    |-  ( ph  ->  ( ps  ->  -.  ch )
 )   =>    |-  ( ph  ->  -.  ps )
 
Theoremnotnoti 619 Infer double negation. (Contributed by NM, 27-Feb-2008.)
 |-  ph   =>    |- 
 -.  -.  ph
 
Theorempm2.21i 620 A contradiction implies anything. Inference from pm2.21 591. (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |- 
 -.  ph   =>    |-  ( ph  ->  ps )
 
Theorempm2.24ii 621 A contradiction implies anything. Inference from pm2.24 595. (Contributed by NM, 27-Feb-2008.)
 |-  ph   &    |- 
 -.  ph   =>    |- 
 ps
 
Theoremnsyld 622 A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
 |-  ( ph  ->  ( ps  ->  -.  ch )
 )   &    |-  ( ph  ->  ( ta  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  -.  ta )
 )
 
Theoremnsyli 623 A negated syllogism inference. (Contributed by NM, 3-May-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  -.  ch )   =>    |-  ( ph  ->  ( th  ->  -. 
 ps ) )
 
Theoremmth8 624 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 625 Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  -.  ( ps  ->  -.  ch )
 )
 
Theoremjcn 626 Inference joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  -.  ch )   =>    |-  ( ph  ->  -.  ( ps  ->  ch ) )
 
Theoremconax1 627 Contrapositive of ax-1 6. (Contributed by BJ, 28-Oct-2023.)
 |-  ( -.  ( ph  ->  ps )  ->  -.  ps )
 
Theoremconax1k 628 Weakening of conax1 627. General instance of pm2.51 629 and of pm2.52 630. (Contributed by BJ, 28-Oct-2023.)
 |-  ( -.  ( ph  ->  ps )  ->  ( ch  ->  -.  ps )
 )
 
Theorempm2.51 629 Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  ->  ps )  ->  ( ph  ->  -.  ps )
 )
 
Theorempm2.52 630 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 631 Exportation theorem pm3.3 259 (closed form of ex 114) expressed with primitive connectives. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( -.  ( ph  ->  -.  ps )  ->  ch )  ->  ( ph  ->  ( ps  ->  ch ) ) )
 
Theoremjarl 632 Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.)
 |-  ( ( ( ph  ->  ps )  ->  ch )  ->  ( -.  ph  ->  ch ) )
 
Theorempm2.65 633 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 851, 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 634 Deduction 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 635 Deduction for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 ps )  ->  -.  ch )   =>    |-  ( ph  ->  -.  ps )
 
Theoremmto 636 The rule of modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
 |- 
 -.  ps   &    |-  ( ph  ->  ps )   =>    |- 
 -.  ph
 
Theoremmtod 637 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 638 Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
 |- 
 -.  ch   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  -.  ps )
 
Theoremmtand 639 A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ph  ->  -.  ps )
 
Theoremnotbi 640 Equivalence property for negation. Closed form. (Contributed by BJ, 27-Jan-2020.)
 |-  ( ( ph  <->  ps )  ->  ( -.  ph  <->  -.  ps ) )
 
Theoremnotbid 641 Equivalence property for negation. Deduction form. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( -.  ps  <->  -.  ch ) )
 
Theoremnotbii 642 Equivalence property for negation. Inference form. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  <->  ps )   =>    |-  ( -.  ph  <->  -.  ps )
 
Theoremcon2b 643 Contraposition. Bidirectional version of con2 617. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  -. 
 ps )  <->  ( ps  ->  -.  ph ) )
 
Theoremmtbi 644 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 645 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 646 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  -. 
 ch )
 
Theoremmtbird 647 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremmtbii 648 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
 |- 
 -.  ps   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  -. 
 ch )
 
Theoremmtbiri 649 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
 |- 
 -.  ch   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremsylnib 650 A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ps  <->  ch )   =>    |-  ( ph  ->  -.  ch )
 
Theoremsylnibr 651 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 652 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 653 A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
 |-  ( ps  <->  ph )   &    |-  ( -.  ps  ->  ch )   =>    |-  ( -.  ph  ->  ch )
 
Theoremxchnxbi 654 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( -.  ph  <->  ps )   &    |-  ( ph  <->  ch )   =>    |-  ( -.  ch  <->  ps )
 
Theoremxchnxbir 655 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( -.  ph  <->  ps )   &    |-  ( ch  <->  ph )   =>    |-  ( -.  ch  <->  ps )
 
Theoremxchbinx 656 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( ph  <->  -.  ps )   &    |-  ( ps 
 <->  ch )   =>    |-  ( ph  <->  -.  ch )
 
Theoremxchbinxr 657 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( ph  <->  -.  ps )   &    |-  ( ch 
 <->  ps )   =>    |-  ( ph  <->  -.  ch )
 
Theoremmt2bi 658 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 659 Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( -.  ph  ->  ( -.  ps  <->  ( ps  ->  ph ) ) )
 
Theoremannimim 660 Express conjunction in terms of implication. One direction of Theorem *4.61 of [WhiteheadRussell] p. 120. The converse holds for decidable propositions, as can be seen at annimdc 906. (Contributed by Jim Kingdon, 24-Dec-2017.)
 |-  ( ( ph  /\  -.  ps )  ->  -.  ( ph  ->  ps ) )
 
Theorempm4.65r 661 One direction of Theorem *4.65 of [WhiteheadRussell] p. 120. The converse holds in classical logic. (Contributed by Jim Kingdon, 28-Jul-2018.)
 |-  ( ( -.  ph  /\ 
 -.  ps )  ->  -.  ( -.  ph  ->  ps )
 )
 
Theoremimanim 662 Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 859. (Contributed by Jim Kingdon, 24-Dec-2017.)
 |-  ( ( ph  ->  ps )  ->  -.  ( ph  /\  -.  ps )
 )
 
Theorempm3.37 663 Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  /\ 
 ps )  ->  ch )  ->  ( ( ph  /\  -.  ch )  ->  -.  ps )
 )
 
Theoremimnan 664 Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.)
 |-  ( ( ph  ->  -. 
 ps )  <->  -.  ( ph  /\  ps ) )
 
Theoremimnani 665 Express implication in terms of conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
 |- 
 -.  ( ph  /\  ps )   =>    |-  ( ph  ->  -.  ps )
 
Theoremnan 666 Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
 |-  ( ( ph  ->  -.  ( ps  /\  ch ) )  <->  ( ( ph  /\ 
 ps )  ->  -.  ch ) )
 
Theorempm3.24 667 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 )
 
Theorempm4.15 668 Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
 |-  ( ( ( ph  /\ 
 ps )  ->  -.  ch ) 
 <->  ( ( ps  /\  ch )  ->  -.  ph )
 )
 
Theorempm5.21 669 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 670 Two propositions are equivalent if they are both false. Closed form of 2false 675. Equivalent to a bi2 129-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 671 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 672 Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
 |-  ( -.  ps  ->  ( ( ph  <->  ps )  <->  -.  ph ) )
 
Theoremnbn 673 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 674 Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
 |-  ph   =>    |-  ( -.  ps  <->  ( ps  <->  -.  ph ) )
 
Theorem2false 675 Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |- 
 -.  ph   &    |-  -.  ps   =>    |-  ( ph  <->  ps )
 
Theorem2falsed 676 Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ph  ->  -.  ch )   =>    |-  ( ph  ->  ( ps 
 <->  ch ) )
 
Theorempm5.21ni 677 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 678 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 679 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 ) )
 
Theorempm5.19 680 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |- 
 -.  ( ph  <->  -.  ph )
 
Theorempm4.8 681 Theorem *4.8 of [WhiteheadRussell] p. 122. This one holds for all propositions, but compare with pm4.81dc 878 which requires a decidability condition. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  -.  ph )  <->  -.  ph )
 
1.2.6  Logical disjunction
 
Syntaxwo 682 Extend wff definition to include disjunction ('or').
 wff  ( ph  \/  ps )
 
Axiomax-io 683 Definition of 'or'. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias jaob 684 instead. (New usage is discouraged.)
 |-  ( ( ( ph  \/  ch )  ->  ps )  <->  ( ( ph  ->  ps )  /\  ( ch  ->  ps )
 ) )
 
Theoremjaob 684 Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. Alias of ax-io 683. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( ( ph  \/  ch )  ->  ps )  <->  ( ( ph  ->  ps )  /\  ( ch  ->  ps )
 ) )
 
Theoremolc 685 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 686 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 ) )
 
Theorempm2.67-2 687 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 688 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 ) )
 
Theorempm3.44 689 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 690 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 691 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 692 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 693 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 694 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 695 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 867. (Contributed by Jim Kingdon, 21-Jul-2018.)
 |-  ( ( -.  ph  \/  ps )  ->  ( ph  ->  ps ) )
 
Theorempm2.53 696 Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 861). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.)
 |-  ( ( ph  \/  ps )  ->  ( -.  ph 
 ->  ps ) )
 
Theoremori 697 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  \/  ps )   =>    |-  ( -.  ph  ->  ps )
 
Theoremord 698 Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ph  ->  ( -.  ps 
 ->  ch ) )
 
Theoremorel1 699 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 700 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 ) )
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