P. 7 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 601-700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	biadani 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 ) ) )
Theorem	biadanii 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)
Axiom	ax-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 )
Axiom	ax-in2 604	'Not' elimination. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
|- ( -. ph -> ( ph -> ps ) )
Theorem	pm2.01 605	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 )
Theorem	pm2.21 606	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 ) )
Theorem	pm2.01d 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 )
Theorem	pm2.21d 608	A contradiction implies anything. Deduction from pm2.21 606. (Contributed by NM, 10-Feb-1996.)
|- ( ph -> -. ps )   =>    |- ( ph -> ( ps -> ch ) )
Theorem	pm2.21dd 609	A contradiction implies anything. Deduction from pm2.21 606. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ph -> ps )   &    |- ( ph -> -. ps )   =>    |- ( ph -> ch )
Theorem	pm2.24 610	Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
|- ( ph -> ( -. ph -> ps ) )
Theorem	pm2.24d 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 ) )
Theorem	pm2.24i 612	Inference version of pm2.24 610. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ph   =>    |- ( -. ph -> ps )
Theorem	con2d 613	A contraposition deduction. (Contributed by NM, 19-Aug-1993.) (Revised by NM, 12-Feb-2013.)
|- ( ph -> ( ps -> -. ch ) )   =>    |- ( ph -> ( ch -> -. ps ) )
Theorem	mt2d 614	Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
|- ( ph -> ch )   &    |- ( ph -> ( ps -> -. ch ) )   =>    |- ( ph -> -. ps )
Theorem	nsyl3 615	A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) (Revised by NM, 13-Jun-2013.)
|- ( ph -> -. ps )   &    |- ( ch -> ps )   =>    |- ( ch -> -. ph )
Theorem	con2i 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 )
Theorem	nsyl 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 )
Theorem	notnot 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 )
Theorem	notnotd 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 )
Theorem	con3d 620	A contraposition deduction. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( -. ch -> -. ps ) )
Theorem	con3i 621	A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
|- ( ph -> ps )   =>    |- ( -. ps -> -. ph )
Theorem	con3rr3 622	Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( -. ch -> ( ph -> -. ps ) )
Theorem	notnotnot 623	Triple negation is equivalent to negation. (Contributed by Jim Kingdon, 28-Jul-2018.)
|- ( -. -. -. ph <-> -. ph )
Theorem	con3dimp 624	Variant of con3d 620 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ( ph /\ -. ch ) -> -. ps )
Theorem	pm2.01da 625	Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ( ph /\ ps ) -> -. ps )   =>    |- ( ph -> -. ps )
Theorem	pm3.2im 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 ) ) )
Theorem	expi 627	An exportation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.)
|- ( -. ( ph -> -. ps ) -> ch )   =>    |- ( ph -> ( ps -> ch ) )
Theorem	pm2.65i 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
Theorem	mt2 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
Theorem	biijust 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 ) ) ) )
Theorem	con3 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 ) )
Theorem	con2 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 ) )
Theorem	mt2i 633	Modus tollens inference. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
|- ch   &    |- ( ph -> ( ps -> -. ch ) )   =>    |- ( ph -> -. ps )
Theorem	notnoti 634	Infer double negation. (Contributed by NM, 27-Feb-2008.)
|- ph   =>    |- -. -. ph
Theorem	pm2.21i 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 )
Theorem	pm2.24ii 636	A contradiction implies anything. Inference from pm2.24 610. (Contributed by NM, 27-Feb-2008.)
|- ph   &    |- -. ph   =>    |- ps
Theorem	nsyld 637	A negated syllogism deduction. (Contributed by NM, 9-Apr-2005.)
|- ( ph -> ( ps -> -. ch ) )   &    |- ( ph -> ( ta -> ch ) )   =>    |- ( ph -> ( ps -> -. ta ) )
Theorem	nsyli 638	A negated syllogism inference. (Contributed by NM, 3-May-1994.)
|- ( ph -> ( ps -> ch ) )   &    |- ( th -> -. ch )   =>    |- ( ph -> ( th -> -. ps ) )
Theorem	jc 639	Inference joining the consequents of two premises. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ps )   &    |- ( ph -> ch )   =>    |- ( ph -> -. ( ps -> -. ch ) )
Theorem	jcn 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 ) ) )
Theorem	jcnd 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 ) )
Theorem	conax1 642	Contrapositive of ax-1 6. (Contributed by BJ, 28-Oct-2023.)
|- ( -. ( ph -> ps ) -> -. ps )
Theorem	conax1k 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 ) )
Theorem	pm2.51 644	Theorem *2.51 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ( -. ( ph -> ps ) -> ( ph -> -. ps ) )
Theorem	pm2.52 645	Theorem *2.52 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( -. ( ph -> ps ) -> ( -. ph -> -. ps ) )
Theorem	expt 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 ) ) )
Theorem	jarl 647	Elimination of a nested antecedent. (Contributed by Wolf Lammen, 10-May-2013.)
|- ( ( ( ph -> ps ) -> ch ) -> ( -. ph -> ch ) )
Theorem	pm2.65 648	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 ) )
Theorem	pm2.65d 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 )
Theorem	pm2.65da 650	Deduction for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ph /\ ps ) -> -. ch )   =>    |- ( ph -> -. ps )
Theorem	mto 651	The rule of modus tollens. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Sep-2013.)
|- -. ps   &    |- ( ph -> ps )   =>    |- -. ph
Theorem	mtod 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 )
Theorem	mtoi 653	Modus tollens inference. (Contributed by NM, 5-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Sep-2012.)
|- -. ch   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> -. ps )
Theorem	mtand 654	A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
|- ( ph -> -. ch )   &    |- ( ( ph /\ ps ) -> ch )   =>    |- ( ph -> -. ps )
Theorem	notbi 655	Equivalence property for negation. Closed form. (Contributed by BJ, 27-Jan-2020.)
|- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) )
Theorem	notbid 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 ) )
Theorem	notbii 657	Equivalence property for negation. Inference form. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ph <-> ps )   =>    |- ( -. ph <-> -. ps )
Theorem	con2b 658	Contraposition. Bidirectional version of con2 632. (Contributed by NM, 5-Aug-1993.)
|- ( ( ph -> -. ps ) <-> ( ps -> -. ph ) )
Theorem	mtbi 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
Theorem	mtbir 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
Theorem	mtbid 661	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
|- ( ph -> -. ps )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> -. ch )
Theorem	mtbird 662	A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
|- ( ph -> -. ch )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> -. ps )
Theorem	mtbii 663	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
|- -. ps   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> -. ch )
Theorem	mtbiri 664	An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
|- -. ch   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> -. ps )
Theorem	sylnib 665	A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
|- ( ph -> -. ps )   &    |- ( ps <-> ch )   =>    |- ( ph -> -. ch )
Theorem	sylnibr 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 )
Theorem	sylnbi 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 )
Theorem	sylnbir 668	A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
|- ( ps <-> ph )   &    |- ( -. ps -> ch )   =>    |- ( -. ph -> ch )
Theorem	xchnxbi 669	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
|- ( -. ph <-> ps )   &    |- ( ph <-> ch )   =>    |- ( -. ch <-> ps )
Theorem	xchnxbir 670	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
|- ( -. ph <-> ps )   &    |- ( ch <-> ph )   =>    |- ( -. ch <-> ps )
Theorem	xchbinx 671	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
|- ( ph <-> -. ps )   &    |- ( ps <-> ch )   =>    |- ( ph <-> -. ch )
Theorem	xchbinxr 672	Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
|- ( ph <-> -. ps )   &    |- ( ch <-> ps )   =>    |- ( ph <-> -. ch )
Theorem	mt2bi 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 ) )
Theorem	mtt 674	Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( -. ph -> ( -. ps <-> ( ps -> ph ) ) )
Theorem	annimim 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 ) )
Theorem	pm4.65r 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 ) )
Theorem	imanim 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 ) )
Theorem	pm3.37 678	Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ph /\ ps ) -> ch ) -> ( ( ph /\ -. ch ) -> -. ps ) )
Theorem	imnan 679	Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.)
|- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
Theorem	imnani 680	Express implication in terms of conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
|- -. ( ph /\ ps )   =>    |- ( ph -> -. ps )
Theorem	nan 681	Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
|- ( ( ph -> -. ( ps /\ ch ) ) <-> ( ( ph /\ ps ) -> -. ch ) )
Theorem	pm3.24 682	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 )
Theorem	pm4.15 683	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 ) )
Theorem	pm5.21 684	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 ) )
Theorem	pm5.21im 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 ) ) )
Theorem	nbn2 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 ) ) )
Theorem	bibif 687	Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)
|- ( -. ps -> ( ( ph <-> ps ) <-> -. ph ) )
Theorem	nbn 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 ) )
Theorem	nbn3 689	Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)
|- ph   =>    |- ( -. ps <-> ( ps <-> -. ph ) )
Theorem	2false 690	Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- -. ph   &    |- -. ps   =>    |- ( ph <-> ps )
Theorem	2falsed 691	Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.)
|- ( ph -> -. ps )   &    |- ( ph -> -. ch )   =>    |- ( ph -> ( ps <-> ch ) )
Theorem	pm5.21ni 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 ) )
Theorem	pm5.21nii 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 )
Theorem	pm5.21ndd 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 ) )
Theorem	pm5.19 695	Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- -. ( ph <-> -. ph )
Theorem	pm4.8 696	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
Syntax	wo 697	Extend wff definition to include disjunction ('or').
wff ( ph \/ ps )
Axiom	ax-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 ) ) )
Theorem	jaob 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 ) ) )
Theorem	olc 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 ) )