P. 7 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 601-700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	imp5a 601	An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
|- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) )   =>    |- ( ph -> ( ps -> ( ch -> ( ( th /\ ta ) -> et ) ) ) )
Theorem	imp5d 602	An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
|- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) )   =>    |- ( ( ( ph /\ ps ) /\ ch ) -> ( ( th /\ ta ) -> et ) )
Theorem	imp5g 603	An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
|- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) )   =>    |- ( ( ph /\ ps ) -> ( ( ( ch /\ th ) /\ ta ) -> et ) )
Theorem	imp55 604	An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
|- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) )   =>    |- ( ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) /\ ta ) -> et )
Theorem	imp511 605	An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
|- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) )   =>    |- ( ( ph /\ ( ( ps /\ ( ch /\ th ) ) /\ ta ) ) -> et )
Theorem	expimpd 606	Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.)
|- ( ( ph /\ ps ) -> ( ch -> th ) )   =>    |- ( ph -> ( ( ps /\ ch ) -> th ) )
Theorem	exp31 607	An exportation inference. (Contributed by NM, 26-Apr-1994.)
|- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ph -> ( ps -> ( ch -> th ) ) )
Theorem	exp32 608	An exportation inference. (Contributed by NM, 26-Apr-1994.)
|- ( ( ph /\ ( ps /\ ch ) ) -> th )   =>    |- ( ph -> ( ps -> ( ch -> th ) ) )
Theorem	exp4a 609	An exportation inference. (Contributed by NM, 26-Apr-1994.)
|- ( ph -> ( ps -> ( ( ch /\ th ) -> ta ) ) )   =>    |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) )
Theorem	exp4b 610	An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
|- ( ( ph /\ ps ) -> ( ( ch /\ th ) -> ta ) )   =>    |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) )
Theorem	exp4c 611	An exportation inference. (Contributed by NM, 26-Apr-1994.)
|- ( ph -> ( ( ( ps /\ ch ) /\ th ) -> ta ) )   =>    |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) )
Theorem	exp4d 612	An exportation inference. (Contributed by NM, 26-Apr-1994.)
|- ( ph -> ( ( ps /\ ( ch /\ th ) ) -> ta ) )   =>    |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) )
Theorem	exp41 613	An exportation inference. (Contributed by NM, 26-Apr-1994.)
|- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta )   =>    |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) )
Theorem	exp42 614	An exportation inference. (Contributed by NM, 26-Apr-1994.)
|- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ta )   =>    |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) )
Theorem	exp43 615	An exportation inference. (Contributed by NM, 26-Apr-1994.)
|- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta )   =>    |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) )
Theorem	exp44 616	An exportation inference. (Contributed by NM, 26-Apr-1994.)
|- ( ( ph /\ ( ( ps /\ ch ) /\ th ) ) -> ta )   =>    |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) )
Theorem	exp45 617	An exportation inference. (Contributed by NM, 26-Apr-1994.)
|- ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) -> ta )   =>    |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) )
Theorem	expr 618	Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
|- ( ( ph /\ ( ps /\ ch ) ) -> th )   =>    |- ( ( ph /\ ps ) -> ( ch -> th ) )
Theorem	exp5c 619	An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
|- ( ph -> ( ( ps /\ ch ) -> ( ( th /\ ta ) -> et ) ) )   =>    |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) )
Theorem	exp5j 620	An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
|- ( ph -> ( ( ( ( ps /\ ch ) /\ th ) /\ ta ) -> et ) )   =>    |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) )
Theorem	exp53 621	An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)
|- ( ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) /\ ta ) -> et )   =>    |- ( ph -> ( ps -> ( ch -> ( th -> ( ta -> et ) ) ) ) )
Theorem	expl 622	Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.)
|- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ph -> ( ( ps /\ ch ) -> th ) )
Theorem	impr 623	Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
|- ( ( ph /\ ps ) -> ( ch -> th ) )   =>    |- ( ( ph /\ ( ps /\ ch ) ) -> th )
Theorem	impl 624	Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.)
|- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ( ( ph /\ ps ) /\ ch ) -> th )
Theorem	impac 625	Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ( ph /\ ps ) -> ( ch /\ ps ) )
Theorem	exbiri 626	Inference form of exbir 36474. This proof is exbiriVD 36894 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.)
|- ( ( ph /\ ps ) -> ( ch <-> th ) )   =>    |- ( ph -> ( ps -> ( th -> ch ) ) )
Theorem	simprbda 627	Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
|- ( ph -> ( ps <-> ( ch /\ th ) ) )   =>    |- ( ( ph /\ ps ) -> ch )
Theorem	simplbda 628	Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
|- ( ph -> ( ps <-> ( ch /\ th ) ) )   =>    |- ( ( ph /\ ps ) -> th )
Theorem	simplbi2 629	Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.)
|- ( ph <-> ( ps /\ ch ) )   =>    |- ( ps -> ( ch -> ph ) )
Theorem	simplbi2comt 630	Closed form of simplbi2com 631. (Contributed by Alan Sare, 22-Jul-2012.)
|- ( ( ph <-> ( ps /\ ch ) ) -> ( ch -> ( ps -> ph ) ) )
Theorem	simplbi2com 631	A deduction eliminating a conjunct, similar to simplbi2 629. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)
|- ( ph <-> ( ps /\ ch ) )   =>    |- ( ch -> ( ps -> ph ) )
Theorem	dfbi2 632	A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49. (Contributed by NM, 24-Jan-1993.)
|- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
Theorem	dfbi 633	Definition df-bi 188 rewritten in an abbreviated form to help intuitive understanding of that definition. Note that it is a conjunction of two implications; one which asserts properties that follow from the biconditional and one which asserts properties that imply the biconditional. (Contributed by NM, 15-Aug-2008.)
|- ( ( ( ph <-> ps ) -> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) /\ ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ( ph <-> ps ) ) )
Theorem	pm4.71 634	Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 2-Dec-2012.)
|- ( ( ph -> ps ) <-> ( ph <-> ( ph /\ ps ) ) )
Theorem	pm4.71r 635	Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 25-Jul-1999.)
|- ( ( ph -> ps ) <-> ( ph <-> ( ps /\ ph ) ) )
Theorem	pm4.71i 636	Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 4-Jan-2004.)
|- ( ph -> ps )   =>    |- ( ph <-> ( ph /\ ps ) )
Theorem	pm4.71ri 637	Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 1-Dec-2003.)
|- ( ph -> ps )   =>    |- ( ph <-> ( ps /\ ph ) )
Theorem	pm4.71d 638	Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by Mario Carneiro, 25-Dec-2016.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ps <-> ( ps /\ ch ) ) )
Theorem	pm4.71rd 639	Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 10-Feb-2005.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ps <-> ( ch /\ ps ) ) )
Theorem	pm5.32 640	Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.)
|- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )
Theorem	pm5.32i 641	Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ( ph /\ ps ) <-> ( ph /\ ch ) )
Theorem	pm5.32ri 642	Distribution of implication over biconditional (inference rule). (Contributed by NM, 12-Mar-1995.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ( ps /\ ph ) <-> ( ch /\ ph ) )
Theorem	pm5.32d 643	Distribution of implication over biconditional (deduction rule). (Contributed by NM, 29-Oct-1996.)
|- ( ph -> ( ps -> ( ch <-> th ) ) )   =>    |- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) )
Theorem	pm5.32rd 644	Distribution of implication over biconditional (deduction rule). (Contributed by NM, 25-Dec-2004.)
|- ( ph -> ( ps -> ( ch <-> th ) ) )   =>    |- ( ph -> ( ( ch /\ ps ) <-> ( th /\ ps ) ) )
Theorem	pm5.32da 645	Distribution of implication over biconditional (deduction rule). (Contributed by NM, 9-Dec-2006.)
|- ( ( ph /\ ps ) -> ( ch <-> th ) )   =>    |- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) )
Theorem	biadan2 646	Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.)
|- ( ph -> ps )   &    |- ( ps -> ( ph <-> ch ) )   =>    |- ( ph <-> ( ps /\ ch ) )
Theorem	pm4.24 647	Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.)
|- ( ph <-> ( ph /\ ph ) )
Theorem	anidm 648	Idempotent law for conjunction. (Contributed by NM, 8-Jan-2004.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)
|- ( ( ph /\ ph ) <-> ph )
Theorem	anidms 649	Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.)
|- ( ( ph /\ ph ) -> ps )   =>    |- ( ph -> ps )
Theorem	anidmdbi 650	Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)
|- ( ( ph -> ( ps /\ ps ) ) <-> ( ph -> ps ) )
Theorem	anasss 651	Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
|- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ( ph /\ ( ps /\ ch ) ) -> th )
Theorem	anassrs 652	Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
|- ( ( ph /\ ( ps /\ ch ) ) -> th )   =>    |- ( ( ( ph /\ ps ) /\ ch ) -> th )
Theorem	anass 653	Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell] p. 118. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
|- ( ( ( ph /\ ps ) /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) )
Theorem	sylanl1 654	A syllogism inference. (Contributed by NM, 10-Mar-2005.)
|- ( ph -> ps )   &    |- ( ( ( ps /\ ch ) /\ th ) -> ta )   =>    |- ( ( ( ph /\ ch ) /\ th ) -> ta )
Theorem	sylanl2 655	A syllogism inference. (Contributed by NM, 1-Jan-2005.)
|- ( ph -> ch )   &    |- ( ( ( ps /\ ch ) /\ th ) -> ta )   =>    |- ( ( ( ps /\ ph ) /\ th ) -> ta )
Theorem	sylanr1 656	A syllogism inference. (Contributed by NM, 9-Apr-2005.)
|- ( ph -> ch )   &    |- ( ( ps /\ ( ch /\ th ) ) -> ta )   =>    |- ( ( ps /\ ( ph /\ th ) ) -> ta )
Theorem	sylanr2 657	A syllogism inference. (Contributed by NM, 9-Apr-2005.)
|- ( ph -> th )   &    |- ( ( ps /\ ( ch /\ th ) ) -> ta )   =>    |- ( ( ps /\ ( ch /\ ph ) ) -> ta )
Theorem	sylani 658	A syllogism inference. (Contributed by NM, 2-May-1996.)
|- ( ph -> ch )   &    |- ( ps -> ( ( ch /\ th ) -> ta ) )   =>    |- ( ps -> ( ( ph /\ th ) -> ta ) )
Theorem	sylan2i 659	A syllogism inference. (Contributed by NM, 1-Aug-1994.)
|- ( ph -> th )   &    |- ( ps -> ( ( ch /\ th ) -> ta ) )   =>    |- ( ps -> ( ( ch /\ ph ) -> ta ) )
Theorem	syl2ani 660	A syllogism inference. (Contributed by NM, 3-Aug-1999.)
|- ( ph -> ch )   &    |- ( et -> th )   &    |- ( ps -> ( ( ch /\ th ) -> ta ) )   =>    |- ( ps -> ( ( ph /\ et ) -> ta ) )
Theorem	sylan9 661	Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)
|- ( ph -> ( ps -> ch ) )   &    |- ( th -> ( ch -> ta ) )   =>    |- ( ( ph /\ th ) -> ( ps -> ta ) )
Theorem	sylan9r 662	Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993.)
|- ( ph -> ( ps -> ch ) )   &    |- ( th -> ( ch -> ta ) )   =>    |- ( ( th /\ ph ) -> ( ps -> ta ) )
Theorem	mtand 663	A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
|- ( ph -> -. ch )   &    |- ( ( ph /\ ps ) -> ch )   =>    |- ( ph -> -. ps )
Theorem	mtord 664	A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.)
|- ( ph -> -. ch )   &    |- ( ph -> -. th )   &    |- ( ph -> ( ps -> ( ch \/ th ) ) )   =>    |- ( ph -> -. ps )
Theorem	syl2anc 665	Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.)
|- ( ph -> ps )   &    |- ( ph -> ch )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ph -> th )
Theorem	sylancl 666	Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ph -> ps )   &    |- ch   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ph -> th )
Theorem	sylancr 667	Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ps   &    |- ( ph -> ch )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ph -> th )
Theorem	sylanbrc 668	Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ph -> ps )   &    |- ( ph -> ch )   &    |- ( th <-> ( ps /\ ch ) )   =>    |- ( ph -> th )
Theorem	sylancb 669	A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
|- ( ph <-> ps )   &    |- ( ph <-> ch )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ph -> th )
Theorem	sylancbr 670	A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
|- ( ps <-> ph )   &    |- ( ch <-> ph )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ph -> th )
Theorem	sylancom 671	Syllogism inference with commutation of antecedents. (Contributed by NM, 2-Jul-2008.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ch /\ ps ) -> th )   =>    |- ( ( ph /\ ps ) -> th )
Theorem	mpdan 672	An inference based on modus ponens. (Contributed by NM, 23-May-1999.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
|- ( ph -> ps )   &    |- ( ( ph /\ ps ) -> ch )   =>    |- ( ph -> ch )
Theorem	mpancom 673	An inference based on modus ponens with commutation of antecedents. (Contributed by NM, 28-Oct-2003.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
|- ( ps -> ph )   &    |- ( ( ph /\ ps ) -> ch )   =>    |- ( ps -> ch )
Theorem	mpan 674	An inference based on modus ponens. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
|- ph   &    |- ( ( ph /\ ps ) -> ch )   =>    |- ( ps -> ch )
Theorem	mpan2 675	An inference based on modus ponens. (Contributed by NM, 16-Sep-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
|- ps   &    |- ( ( ph /\ ps ) -> ch )   =>    |- ( ph -> ch )
Theorem	mp2an 676	An inference based on modus ponens. (Contributed by NM, 13-Apr-1995.)
|- ph   &    |- ps   &    |- ( ( ph /\ ps ) -> ch )   =>    |- ch
Theorem	mp4an 677	An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2010.)
|- ph   &    |- ps   &    |- ch   &    |- th   &    |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta )   =>    |- ta
Theorem	mpan2d 678	A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
|- ( ph -> ch )   &    |- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ph -> ( ps -> th ) )
Theorem	mpand 679	A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
|- ( ph -> ps )   &    |- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ph -> ( ch -> th ) )
Theorem	mpani 680	An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
|- ps   &    |- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ph -> ( ch -> th ) )
Theorem	mpan2i 681	An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
|- ch   &    |- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ph -> ( ps -> th ) )
Theorem	mp2ani 682	An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)
|- ps   &    |- ch   &    |- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ph -> th )
Theorem	mp2and 683	A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
|- ( ph -> ps )   &    |- ( ph -> ch )   &    |- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ph -> th )
Theorem	mpanl1 684	An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
|- ph   &    |- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ( ps /\ ch ) -> th )
Theorem	mpanl2 685	An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)
|- ps   &    |- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ( ph /\ ch ) -> th )
Theorem	mpanl12 686	An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
|- ph   &    |- ps   &    |- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ch -> th )
Theorem	mpanr1 687	An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)
|- ps   &    |- ( ( ph /\ ( ps /\ ch ) ) -> th )   =>    |- ( ( ph /\ ch ) -> th )
Theorem	mpanr2 688	An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
|- ch   &    |- ( ( ph /\ ( ps /\ ch ) ) -> th )   =>    |- ( ( ph /\ ps ) -> th )
Theorem	mpanr12 689	An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.)
|- ps   &    |- ch   &    |- ( ( ph /\ ( ps /\ ch ) ) -> th )   =>    |- ( ph -> th )
Theorem	mpanlr1 690	An inference based on modus ponens. (Contributed by NM, 30-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
|- ps   &    |- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ta )   =>    |- ( ( ( ph /\ ch ) /\ th ) -> ta )
Theorem	pm5.74da 691	Distribution of implication over biconditional (deduction rule). (Contributed by NM, 4-May-2007.)
|- ( ( ph /\ ps ) -> ( ch <-> th ) )   =>    |- ( ph -> ( ( ps -> ch ) <-> ( ps -> th ) ) )
Theorem	pm4.45 692	Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
|- ( ph <-> ( ph /\ ( ph \/ ps ) ) )
Theorem	imdistan 693	Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.)
|- ( ( ph -> ( ps -> ch ) ) <-> ( ( ph /\ ps ) -> ( ph /\ ch ) ) )
Theorem	imdistani 694	Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ( ph /\ ps ) -> ( ph /\ ch ) )
Theorem	imdistanri 695	Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ( ps /\ ph ) -> ( ch /\ ph ) )
Theorem	imdistand 696	Distribution of implication with conjunction (deduction rule). (Contributed by NM, 27-Aug-2004.)
|- ( ph -> ( ps -> ( ch -> th ) ) )   =>    |- ( ph -> ( ( ps /\ ch ) -> ( ps /\ th ) ) )
Theorem	imdistanda 697	Distribution of implication with conjunction (deduction version with conjoined antecedent). (Contributed by Jeff Madsen, 19-Jun-2011.)
|- ( ( ph /\ ps ) -> ( ch -> th ) )   =>    |- ( ph -> ( ( ps /\ ch ) -> ( ps /\ th ) ) )
Theorem	anbi2i 698	Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
|- ( ph <-> ps )   =>    |- ( ( ch /\ ph ) <-> ( ch /\ ps ) )
Theorem	anbi1i 699	Introduce a right conjunct to both sides of a logical equivalence. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
|- ( ph <-> ps )   =>    |- ( ( ph /\ ch ) <-> ( ps /\ ch ) )
Theorem	anbi2ci 700	Variant of anbi2i 698 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( ph <-> ps )   =>    |- ( ( ph /\ ch ) <-> ( ch /\ ps ) )