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Theorem List for Intuitionistic Logic Explorer - 601-700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembiadani 601 An implication implies to the equivalence of some implied equivalence and some other equivalence involving a conjunction. (Contributed by BJ, 4-Mar-2023.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ps  ->  ( ph 
 <->  ch ) )  <->  ( ph  <->  ( ps  /\  ch ) ) )
 
Theorembiadanii 602 Inference associated with biadani 601. Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.) (Proof shortened by BJ, 4-Mar-2023.)
 |-  ( ph  ->  ps )   &    |-  ( ps  ->  ( ph  <->  ch ) )   =>    |-  ( ph  <->  ( ps  /\  ch ) )
 
1.2.5  Logical negation (intuitionistic)
 
Axiomax-in1 603 'Not' introduction. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias pm2.01 605 instead. (New usage is discouraged.)
 |-  ( ( ph  ->  -.  ph )  ->  -.  ph )
 
Axiomax-in2 604 'Not' elimination. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
 |-  ( -.  ph  ->  (
 ph  ->  ps ) )
 
Theorempm2.01 605 Reductio ad absurdum. Theorem *2.01 of [WhiteheadRussell] p. 100. This is valid intuitionistically (in the terminology of Section 1.2 of [Bauer] p. 482 it is a proof of negation not a proof by contradiction); compare with pm2.18dc 840 which only holds for some propositions. Also called weak Clavius law. (Contributed by Mario Carneiro, 12-May-2015.)
 |-  ( ( ph  ->  -.  ph )  ->  -.  ph )
 
Theorempm2.21 606 From a wff and its negation, anything is true. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. (Contributed by Mario Carneiro, 12-May-2015.)
 |-  ( -.  ph  ->  (
 ph  ->  ps ) )
 
Theorempm2.01d 607 Deduction based on reductio ad absurdum. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  ->  ( ps  ->  -.  ps )
 )   =>    |-  ( ph  ->  -.  ps )
 
Theorempm2.21d 608 A contradiction implies anything. Deduction from pm2.21 606. (Contributed by NM, 10-Feb-1996.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theorempm2.21dd 609 A contradiction implies anything. Deduction from pm2.21 606. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  ch )
 
Theorempm2.24 610 Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  ->  ( -.  ph  ->  ps )
 )
 
Theorempm2.24d 611 Deduction version of pm2.24 610. (Contributed by NM, 30-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( -.  ps  ->  ch ) )
 
Theorempm2.24i 612 Inference version of pm2.24 610. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ph   =>    |-  ( -.  ph  ->  ps )
 
Theoremcon2d 613 A contraposition deduction. (Contributed by NM, 19-Aug-1993.) (Revised by NM, 12-Feb-2013.)
 |-  ( ph  ->  ( ps  ->  -.  ch )
 )   =>    |-  ( ph  ->  ( ch  ->  -.  ps )
 )
 
Theoremmt2d 614 Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
 |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ps  ->  -. 
 ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremnsyl3 615 A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) (Revised by NM, 13-Jun-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ch  ->  ps )   =>    |-  ( ch  ->  -.  ph )
 
Theoremcon2i 616 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 617 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 618 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 816) and in particular for decidable propositions (see notnotrdc 828). See also notnotnot 623. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
 |-  ( ph  ->  -.  -.  ph )
 
Theoremnotnotd 619 Deduction associated with notnot 618 and notnoti 634. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  -.  -.  ps )
 
Theoremcon3d 620 A contraposition deduction. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( -.  ch  ->  -.  ps ) )
 
Theoremcon3i 621 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
 |-  ( ph  ->  ps )   =>    |-  ( -.  ps  ->  -.  ph )
 
Theoremcon3rr3 622 Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( -.  ch  ->  ( ph  ->  -.  ps ) )
 
Theoremnotnotnot 623 Triple negation is equivalent to negation. (Contributed by Jim Kingdon, 28-Jul-2018.)
 |-  ( -.  -.  -.  ph  <->  -.  ph )
 
Theoremcon3dimp 624 Variant of con3d 620 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ph  /\ 
 -.  ch )  ->  -.  ps )
 
Theorempm2.01da 625 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ph  /\  ps )  ->  -.  ps )   =>    |-  ( ph  ->  -.  ps )
 
Theorempm3.2im 626 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 627 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 628 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 629 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 630 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 631 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 632 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 633 Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
 |- 
 ch   &    |-  ( ph  ->  ( ps  ->  -.  ch )
 )   =>    |-  ( ph  ->  -.  ps )
 
Theoremnotnoti 634 Infer double negation. (Contributed by NM, 27-Feb-2008.)
 |-  ph   =>    |- 
 -.  -.  ph
 
Theorempm2.21i 635 A contradiction implies anything. Inference from pm2.21 606. (Contributed by NM, 16-Sep-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |- 
 -.  ph   =>    |-  ( ph  ->  ps )
 
Theorempm2.24ii 636 A contradiction implies anything. Inference from pm2.24 610. (Contributed by NM, 27-Feb-2008.)
 |-  ph   &    |- 
 -.  ph   =>    |- 
 ps
 
Theoremnsyld 637 A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
 |-  ( ph  ->  ( ps  ->  -.  ch )
 )   &    |-  ( ph  ->  ( ta  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  -.  ta )
 )
 
Theoremnsyli 638 A negated syllogism inference. (Contributed by NM, 3-May-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  -.  ch )   =>    |-  ( ph  ->  ( th  ->  -. 
 ps ) )
 
Theoremjc 639 Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  -.  ( ps  ->  -.  ch )
 )
 
Theoremjcn 640 Theorem joining the consequents of two premises. Theorem 8 of [Margaris] p. 60. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
 |-  ( ph  ->  ( -.  ps  ->  -.  ( ph  ->  ps ) ) )
 
Theoremjcnd 641 Deduction joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Apr-2024.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  -.  ch )   =>    |-  ( ph  ->  -.  ( ps  ->  ch ) )
 
Theoremconax1 642 Contrapositive of ax-1 6. (Contributed by BJ, 28-Oct-2023.)
 |-  ( -.  ( ph  ->  ps )  ->  -.  ps )
 
Theoremconax1k 643 Weakening of conax1 642. General instance of pm2.51 644 and of pm2.52 645. (Contributed by BJ, 28-Oct-2023.)
 |-  ( -.  ( ph  ->  ps )  ->  ( ch  ->  -.  ps )
 )
 
Theorempm2.51 644 Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
 |-  ( -.  ( ph  ->  ps )  ->  ( ph  ->  -.  ps )
 )
 
Theorempm2.52 645 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 646 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 647 Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.)
 |-  ( ( ( ph  ->  ps )  ->  ch )  ->  ( -.  ph  ->  ch ) )
 
Theorempm2.65 648 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 866, 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 649 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 650 Deduction for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 ps )  ->  -.  ch )   =>    |-  ( ph  ->  -.  ps )
 
Theoremmto 651 The rule of modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
 |- 
 -.  ps   &    |-  ( ph  ->  ps )   =>    |- 
 -.  ph
 
Theoremmtod 652 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 653 Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
 |- 
 -.  ch   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  -.  ps )
 
Theoremmtand 654 A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ph  ->  -.  ps )
 
Theoremnotbi 655 Equivalence property for negation. Closed form. (Contributed by BJ, 27-Jan-2020.)
 |-  ( ( ph  <->  ps )  ->  ( -.  ph  <->  -.  ps ) )
 
Theoremnotbid 656 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 657 Equivalence property for negation. Inference form. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  <->  ps )   =>    |-  ( -.  ph  <->  -.  ps )
 
Theoremcon2b 658 Contraposition. Bidirectional version of con2 632. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  -. 
 ps )  <->  ( ps  ->  -.  ph ) )
 
Theoremmtbi 659 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 660 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 661 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  -. 
 ch )
 
Theoremmtbird 662 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremmtbii 663 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
 |- 
 -.  ps   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  -. 
 ch )
 
Theoremmtbiri 664 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
 |- 
 -.  ch   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremsylnib 665 A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ps  <->  ch )   =>    |-  ( ph  ->  -.  ch )
 
Theoremsylnibr 666 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 667 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 668 A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
 |-  ( ps  <->  ph )   &    |-  ( -.  ps  ->  ch )   =>    |-  ( -.  ph  ->  ch )
 
Theoremxchnxbi 669 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( -.  ph  <->  ps )   &    |-  ( ph  <->  ch )   =>    |-  ( -.  ch  <->  ps )
 
Theoremxchnxbir 670 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( -.  ph  <->  ps )   &    |-  ( ch  <->  ph )   =>    |-  ( -.  ch  <->  ps )
 
Theoremxchbinx 671 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( ph  <->  -.  ps )   &    |-  ( ps 
 <->  ch )   =>    |-  ( ph  <->  -.  ch )
 
Theoremxchbinxr 672 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
 |-  ( ph  <->  -.  ps )   &    |-  ( ch 
 <->  ps )   =>    |-  ( ph  <->  -.  ch )
 
Theoremmt2bi 673 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 674 Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( -.  ph  ->  ( -.  ps  <->  ( ps  ->  ph ) ) )
 
Theoremannimim 675 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 921. (Contributed by Jim Kingdon, 24-Dec-2017.)
 |-  ( ( ph  /\  -.  ps )  ->  -.  ( ph  ->  ps ) )
 
Theorempm4.65r 676 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 677 Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 874. (Contributed by Jim Kingdon, 24-Dec-2017.)
 |-  ( ( ph  ->  ps )  ->  -.  ( ph  /\  -.  ps )
 )
 
Theorempm3.37 678 Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  /\ 
 ps )  ->  ch )  ->  ( ( ph  /\  -.  ch )  ->  -.  ps )
 )
 
Theoremimnan 679 Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.)
 |-  ( ( ph  ->  -. 
 ps )  <->  -.  ( ph  /\  ps ) )
 
Theoremimnani 680 Express implication in terms of conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
 |- 
 -.  ( ph  /\  ps )   =>    |-  ( ph  ->  -.  ps )
 
Theoremnan 681 Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
 |-  ( ( ph  ->  -.  ( ps  /\  ch ) )  <->  ( ( ph  /\ 
 ps )  ->  -.  ch ) )
 
Theorempm3.24 682 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 )
 
Theorempm4.15 683 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 )
 )
 
Theorempm5.21 684 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 685 Two propositions are equivalent if they are both false. Closed form of 2false 690. 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 686 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 687 Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
 |-  ( -.  ps  ->  ( ( ph  <->  ps )  <->  -.  ph ) )
 
Theoremnbn 688 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 689 Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
 |-  ph   =>    |-  ( -.  ps  <->  ( ps  <->  -.  ph ) )
 
Theorem2false 690 Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |- 
 -.  ph   &    |-  -.  ps   =>    |-  ( ph  <->  ps )
 
Theorem2falsed 691 Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ph  ->  -.  ch )   =>    |-  ( ph  ->  ( ps 
 <->  ch ) )
 
Theorempm5.21ni 692 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 693 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 694 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 ) )
 
Theorempm5.19 695 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |- 
 -.  ( ph  <->  -.  ph )
 
Theorempm4.8 696 Theorem *4.8 of [WhiteheadRussell] p. 122. This one holds for all propositions, but compare with pm4.81dc 893 which requires a decidability condition. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  -.  ph )  <->  -.  ph )
 
1.2.6  Logical disjunction
 
Syntaxwo 697 Extend wff definition to include disjunction ('or').
 wff  ( ph  \/  ps )
 
Axiomax-io 698 Definition of 'or'. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias jaob 699 instead. (New usage is discouraged.)
 |-  ( ( ( ph  \/  ch )  ->  ps )  <->  ( ( ph  ->  ps )  /\  ( ch  ->  ps )
 ) )
 
Theoremjaob 699 Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. Alias of ax-io 698. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( ( ph  \/  ch )  ->  ps )  <->  ( ( ph  ->  ps )  /\  ( ch  ->  ps )
 ) )
 
Theoremolc 700 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 ) )
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