P. 5 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 401-500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	syl2ani 401	A syllogism inference. (Contributed by NM, 3-Aug-1999.)
|- ( ph -> ch )   &    |- ( et -> th )   &    |- ( ps -> ( ( ch /\ th ) -> ta ) )   =>    |- ( ps -> ( ( ph /\ et ) -> ta ) )
Theorem	sylan9 402	Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)
|- ( ph -> ( ps -> ch ) )   &    |- ( th -> ( ch -> ta ) )   =>    |- ( ( ph /\ th ) -> ( ps -> ta ) )
Theorem	sylan9r 403	Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps -> ch ) )   &    |- ( th -> ( ch -> ta ) )   =>    |- ( ( th /\ ph ) -> ( ps -> ta ) )
Theorem	syl2anc 404	Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.)
|- ( ph -> ps )   &    |- ( ph -> ch )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ph -> th )
Theorem	sylancl 405	Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ph -> ps )   &    |- ch   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ph -> th )
Theorem	sylancr 406	Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ps   &    |- ( ph -> ch )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ph -> th )
Theorem	sylanblc 407	Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
|- ( ph -> ps )   &    |- ch   &    |- ( ( ps /\ ch ) <-> th )   =>    |- ( ph -> th )
Theorem	sylanblrc 408	Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
|- ( ph -> ps )   &    |- ch   &    |- ( th <-> ( ps /\ ch ) )   =>    |- ( ph -> th )
Theorem	sylanbrc 409	Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ph -> ps )   &    |- ( ph -> ch )   &    |- ( th <-> ( ps /\ ch ) )   =>    |- ( ph -> th )
Theorem	sylancb 410	A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
|- ( ph <-> ps )   &    |- ( ph <-> ch )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ph -> th )
Theorem	sylancbr 411	A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
|- ( ps <-> ph )   &    |- ( ch <-> ph )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ph -> th )
Theorem	sylancom 412	Syllogism inference with commutation of antecents. (Contributed by NM, 2-Jul-2008.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ch /\ ps ) -> th )   =>    |- ( ( ph /\ ps ) -> th )
Theorem	mpdan 413	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 414	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	mpidan 415	A deduction which "stacks" a hypothesis. (Contributed by Stanislas Polu, 9-Mar-2020.) (Proof shortened by Wolf Lammen, 28-Mar-2021.)
|- ( ph -> ch )   &    |- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ( ph /\ ps ) -> th )
Theorem	mpan 416	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 417	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 418	An inference based on modus ponens. (Contributed by NM, 13-Apr-1995.)
|- ph   &    |- ps   &    |- ( ( ph /\ ps ) -> ch )   =>    |- ch
Theorem	mp4an 419	An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2011.)
|- ph   &    |- ps   &    |- ch   &    |- th   &    |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta )   =>    |- ta
Theorem	mpan2d 420	A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
|- ( ph -> ch )   &    |- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ph -> ( ps -> th ) )
Theorem	mpand 421	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 422	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 423	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 424	An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)
|- ps   &    |- ch   &    |- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ph -> th )
Theorem	mp2and 425	A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
|- ( ph -> ps )   &    |- ( ph -> ch )   &    |- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ph -> th )
Theorem	mpanl1 426	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 427	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 428	An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
|- ph   &    |- ps   &    |- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ch -> th )
Theorem	mpanr1 429	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 430	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 431	An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.)
|- ps   &    |- ch   &    |- ( ( ph /\ ( ps /\ ch ) ) -> th )   =>    |- ( ph -> th )
Theorem	mpanlr1 432	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 433	Distribution of implication over biconditional (deduction form). (Contributed by NM, 4-May-2007.)
|- ( ( ph /\ ps ) -> ( ch <-> th ) )   =>    |- ( ph -> ( ( ps -> ch ) <-> ( ps -> th ) ) )
Theorem	imdistan 434	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 435	Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ( ph /\ ps ) -> ( ph /\ ch ) )
Theorem	imdistanri 436	Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ( ps /\ ph ) -> ( ch /\ ph ) )
Theorem	imdistand 437	Distribution of implication with conjunction (deduction form). (Contributed by NM, 27-Aug-2004.)
|- ( ph -> ( ps -> ( ch -> th ) ) )   =>    |- ( ph -> ( ( ps /\ ch ) -> ( ps /\ th ) ) )
Theorem	imdistanda 438	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	pm5.32d 439	Distribution of implication over biconditional (deduction form). (Contributed by NM, 29-Oct-1996.) (Revised by NM, 31-Jan-2015.)
|- ( ph -> ( ps -> ( ch <-> th ) ) )   =>    |- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) )
Theorem	pm5.32rd 440	Distribution of implication over biconditional (deduction form). (Contributed by NM, 25-Dec-2004.)
|- ( ph -> ( ps -> ( ch <-> th ) ) )   =>    |- ( ph -> ( ( ch /\ ps ) <-> ( th /\ ps ) ) )
Theorem	pm5.32da 441	Distribution of implication over biconditional (deduction form). (Contributed by NM, 9-Dec-2006.)
|- ( ( ph /\ ps ) -> ( ch <-> th ) )   =>    |- ( ph -> ( ( ps /\ ch ) <-> ( ps /\ th ) ) )
Theorem	pm5.32 442	Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Revised by NM, 31-Jan-2015.)
|- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )
Theorem	pm5.32i 443	Distribution of implication over biconditional (inference form). (Contributed by NM, 1-Aug-1994.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ( ph /\ ps ) <-> ( ph /\ ch ) )
Theorem	pm5.32ri 444	Distribution of implication over biconditional (inference form). (Contributed by NM, 12-Mar-1995.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ( ps /\ ph ) <-> ( ch /\ ph ) )
Theorem	biadan2 445	Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.)
|- ( ph -> ps )   &    |- ( ps -> ( ph <-> ch ) )   =>    |- ( ph <-> ( ps /\ ch ) )
Theorem	anbi2i 446	Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
|- ( ph <-> ps )   =>    |- ( ( ch /\ ph ) <-> ( ch /\ ps ) )
Theorem	anbi1i 447	Introduce a right conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
|- ( ph <-> ps )   =>    |- ( ( ph /\ ch ) <-> ( ps /\ ch ) )
Theorem	anbi2ci 448	Variant of anbi2i 446 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( ph <-> ps )   =>    |- ( ( ph /\ ch ) <-> ( ch /\ ps ) )
Theorem	anbi12i 449	Conjoin both sides of two equivalences. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   =>    |- ( ( ph /\ ch ) <-> ( ps /\ th ) )
Theorem	anbi12ci 450	Variant of anbi12i 449 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   =>    |- ( ( ph /\ ch ) <-> ( th /\ ps ) )
Theorem	sylan9bb 451	Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( th -> ( ch <-> ta ) )   =>    |- ( ( ph /\ th ) -> ( ps <-> ta ) )
Theorem	sylan9bbr 452	Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( th -> ( ch <-> ta ) )   =>    |- ( ( th /\ ph ) -> ( ps <-> ta ) )
Theorem	anbi2d 453	Deduction adding a left conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( th /\ ps ) <-> ( th /\ ch ) ) )
Theorem	anbi1d 454	Deduction adding a right conjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( ps /\ th ) <-> ( ch /\ th ) ) )
Theorem	anbi1 455	Introduce a right conjunct to both sides of a logical equivalence. Theorem *4.36 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph <-> ps ) -> ( ( ph /\ ch ) <-> ( ps /\ ch ) ) )
Theorem	anbi2 456	Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 16-Nov-2013.)
|- ( ( ph <-> ps ) -> ( ( ch /\ ph ) <-> ( ch /\ ps ) ) )
Theorem	bitr 457	Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ph <-> ps ) /\ ( ps <-> ch ) ) -> ( ph <-> ch ) )
Theorem	anbi12d 458	Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ( ps /\ th ) <-> ( ch /\ ta ) ) )
Theorem	mpan10 459	Modus ponens mixed with several conjunctions. (Contributed by Jim Kingdon, 7-Jan-2018.)
|- ( ( ( ( ph -> ps ) /\ ch ) /\ ph ) -> ( ps /\ ch ) )
Theorem	pm5.3 460	Theorem *5.3 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.)
|- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ph /\ ps ) -> ( ph /\ ch ) ) )
Theorem	adantll 461	Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( th /\ ph ) /\ ps ) -> ch )
Theorem	adantlr 462	Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( ph /\ th ) /\ ps ) -> ch )
Theorem	adantrl 463	Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ph /\ ( th /\ ps ) ) -> ch )
Theorem	adantrr 464	Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ph /\ ( ps /\ th ) ) -> ch )
Theorem	adantlll 465	Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 2-Dec-2012.)
|- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ( ( ( ta /\ ph ) /\ ps ) /\ ch ) -> th )
Theorem	adantllr 466	Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
|- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ( ( ( ph /\ ta ) /\ ps ) /\ ch ) -> th )
Theorem	adantlrl 467	Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
|- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ( ( ph /\ ( ta /\ ps ) ) /\ ch ) -> th )
Theorem	adantlrr 468	Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
|- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ( ( ph /\ ( ps /\ ta ) ) /\ ch ) -> th )
Theorem	adantrll 469	Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
|- ( ( ph /\ ( ps /\ ch ) ) -> th )   =>    |- ( ( ph /\ ( ( ta /\ ps ) /\ ch ) ) -> th )
Theorem	adantrlr 470	Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
|- ( ( ph /\ ( ps /\ ch ) ) -> th )   =>    |- ( ( ph /\ ( ( ps /\ ta ) /\ ch ) ) -> th )
Theorem	adantrrl 471	Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
|- ( ( ph /\ ( ps /\ ch ) ) -> th )   =>    |- ( ( ph /\ ( ps /\ ( ta /\ ch ) ) ) -> th )
Theorem	adantrrr 472	Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
|- ( ( ph /\ ( ps /\ ch ) ) -> th )   =>    |- ( ( ph /\ ( ps /\ ( ch /\ ta ) ) ) -> th )
Theorem	ad2antrr 473	Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.)
|- ( ph -> ps )   =>    |- ( ( ( ph /\ ch ) /\ th ) -> ps )
Theorem	ad2antlr 474	Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.) (Proof shortened by Wolf Lammen, 20-Nov-2012.)
|- ( ph -> ps )   =>    |- ( ( ( ch /\ ph ) /\ th ) -> ps )
Theorem	ad2antrl 475	Deduction adding two conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
|- ( ph -> ps )   =>    |- ( ( ch /\ ( ph /\ th ) ) -> ps )
Theorem	ad2antll 476	Deduction adding conjuncts to antecedent. (Contributed by NM, 19-Oct-1999.)
|- ( ph -> ps )   =>    |- ( ( ch /\ ( th /\ ph ) ) -> ps )
Theorem	ad3antrrr 477	Deduction adding three conjuncts to antecedent. (Contributed by NM, 28-Jul-2012.)
|- ( ph -> ps )   =>    |- ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) -> ps )
Theorem	ad3antlr 478	Deduction adding three conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
|- ( ph -> ps )   =>    |- ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) -> ps )
Theorem	ad4antr 479	Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> ps )   =>    |- ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et ) -> ps )
Theorem	ad4antlr 480	Deduction adding 4 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
|- ( ph -> ps )   =>    |- ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et ) -> ps )
Theorem	ad5antr 481	Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> ps )   =>    |- ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et ) /\ ze ) -> ps )
Theorem	ad5antlr 482	Deduction adding 5 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
|- ( ph -> ps )   =>    |- ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et ) /\ ze ) -> ps )
Theorem	ad6antr 483	Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> ps )   =>    |- ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et ) /\ ze ) /\ si ) -> ps )
Theorem	ad6antlr 484	Deduction adding 6 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
|- ( ph -> ps )   =>    |- ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et ) /\ ze ) /\ si ) -> ps )
Theorem	ad7antr 485	Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> ps )   =>    |- ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et ) /\ ze ) /\ si ) /\ rh ) -> ps )
Theorem	ad7antlr 486	Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
|- ( ph -> ps )   =>    |- ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et ) /\ ze ) /\ si ) /\ rh ) -> ps )
Theorem	ad8antr 487	Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> ps )   =>    |- ( ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) -> ps )
Theorem	ad8antlr 488	Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
|- ( ph -> ps )   =>    |- ( ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) -> ps )
Theorem	ad9antr 489	Deduction adding 9 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> ps )   =>    |- ( ( ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ps )
Theorem	ad9antlr 490	Deduction adding 9 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
|- ( ph -> ps )   =>    |- ( ( ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) -> ps )
Theorem	ad10antr 491	Deduction adding 10 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> ps )   =>    |- ( ( ( ( ( ( ( ( ( ( ( ph /\ ch ) /\ th ) /\ ta ) /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ps )
Theorem	ad10antlr 492	Deduction adding 10 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
|- ( ph -> ps )   =>    |- ( ( ( ( ( ( ( ( ( ( ( ch /\ ph ) /\ th ) /\ ta ) /\ et ) /\ ze ) /\ si ) /\ rh ) /\ mu ) /\ la ) /\ ka ) -> ps )
Theorem	ad2ant2l 493	Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( th /\ ph ) /\ ( ta /\ ps ) ) -> ch )
Theorem	ad2ant2r 494	Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( ph /\ th ) /\ ( ps /\ ta ) ) -> ch )
Theorem	ad2ant2lr 495	Deduction adding two conjuncts to antecedent. (Contributed by NM, 23-Nov-2007.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( th /\ ph ) /\ ( ps /\ ta ) ) -> ch )
Theorem	ad2ant2rl 496	Deduction adding two conjuncts to antecedent. (Contributed by NM, 24-Nov-2007.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( ph /\ th ) /\ ( ta /\ ps ) ) -> ch )
Theorem	simpll 497	Simplification of a conjunction. (Contributed by NM, 18-Mar-2007.)
|- ( ( ( ph /\ ps ) /\ ch ) -> ph )
Theorem	simplr 498	Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.)
|- ( ( ( ph /\ ps ) /\ ch ) -> ps )
Theorem	simprl 499	Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
|- ( ( ph /\ ( ps /\ ch ) ) -> ps )
Theorem	simprr 500	Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
|- ( ( ph /\ ( ps /\ ch ) ) -> ch )