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Theorem List for Metamath Proof Explorer - 601-700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexpl 601 Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )
 
Theoremimpr 602 Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
 |-  ( ( ph  /\  ps )  ->  ( ch  ->  th ) )   =>    |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )
 
Theoremimpl 603 Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ( (
 ph  /\  ps )  /\  ch )  ->  th )
 
Theoremimpac 604 Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( ch  /\  ps ) )
 
Theoremexbiri 605 Inference form of exbir 1355. This proof is exbiriVD 28630 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 ) ) )
 
Theoremsimprbda 606 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
 |-  ( ph  ->  ( ps 
 <->  ( ch  /\  th ) ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremsimplbda 607 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
 |-  ( ph  ->  ( ps 
 <->  ( ch  /\  th ) ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  th )
 
Theoremsimplbi2 608 Deduction eliminating a conjunct. Automatically derived from simplbi2VD 28622. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ps  ->  ( ch  ->  ph ) )
 
Theoremdfbi2 609 A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  <->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph ) ) )
 
Theoremdfbi 610 Definition df-bi 177 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 ) ) )
 
Theorempm4.71 611 Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 2-Dec-2012.)
 |-  ( ( ph  ->  ps )  <->  ( ph  <->  ( ph  /\  ps ) ) )
 
Theorempm4.71r 612 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 ) ) )
 
Theorempm4.71i 613 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 ) )
 
Theorempm4.71ri 614 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 )
 )
 
Theorempm4.71d 615 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 ) ) )
 
Theorempm4.71rd 616 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 ) ) )
 
Theorempm5.32 617 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 ) ) )
 
Theorempm5.32i 618 Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  <->  ( ph  /\  ch ) )
 
Theorempm5.32ri 619 Distribution of implication over biconditional (inference rule). (Contributed by NM, 12-Mar-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ps 
 /\  ph )  <->  ( ch  /\  ph ) )
 
Theorempm5.32d 620 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 29-Oct-1996.)
 |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  <->  ( ps  /\  th ) ) )
 
Theorempm5.32rd 621 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 25-Dec-2004.)
 |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )   =>    |-  ( ph  ->  ( ( ch  /\  ps )  <->  ( th  /\  ps ) ) )
 
Theorempm5.32da 622 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 9-Dec-2006.)
 |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch ) 
 <->  ( ps  /\  th ) ) )
 
Theorembiadan2 623 Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ps  ->  ( ph  <->  ch ) )   =>    |-  ( ph  <->  ( ps  /\  ch ) )
 
Theorempm4.24 624 Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  <->  ( ph  /\  ph )
 )
 
Theoremanidm 625 Idempotent law for conjunction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)
 |-  ( ( ph  /\  ph )  <->  ph )
 
Theoremanidms 626 Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.)
 |-  ( ( ph  /\  ph )  ->  ps )   =>    |-  ( ph  ->  ps )
 
Theoremanidmdbi 627 Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)
 |-  ( ( ph  ->  ( ps  /\  ps )
 ) 
 <->  ( ph  ->  ps )
 )
 
Theoremanasss 628 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )
 
Theoremanassrs 629 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )
 
Theoremanass 630 Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch ) 
 <->  ( ph  /\  ( ps  /\  ch ) ) )
 
Theoremsylanl1 631 A syllogism inference. (Contributed by NM, 10-Mar-2005.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ( ps  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  th )  ->  ta )
 
Theoremsylanl2 632 A syllogism inference. (Contributed by NM, 1-Jan-2005.)
 |-  ( ph  ->  ch )   &    |-  (
 ( ( ps  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( ps 
 /\  ph )  /\  th )  ->  ta )
 
Theoremsylanr1 633 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
 |-  ( ph  ->  ch )   &    |-  (
 ( ps  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ( ps  /\  ( ph  /\  th )
 )  ->  ta )
 
Theoremsylanr2 634 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
 |-  ( ph  ->  th )   &    |-  (
 ( ps  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ( ps  /\  ( ch  /\  ph )
 )  ->  ta )
 
Theoremsylani 635 A syllogism inference. (Contributed by NM, 2-May-1996.)
 |-  ( ph  ->  ch )   &    |-  ( ps  ->  ( ( ch 
 /\  th )  ->  ta )
 )   =>    |-  ( ps  ->  (
 ( ph  /\  th )  ->  ta ) )
 
Theoremsylan2i 636 A syllogism inference. (Contributed by NM, 1-Aug-1994.)
 |-  ( ph  ->  th )   &    |-  ( ps  ->  ( ( ch 
 /\  th )  ->  ta )
 )   =>    |-  ( ps  ->  (
 ( ch  /\  ph )  ->  ta ) )
 
Theoremsyl2ani 637 A syllogism inference. (Contributed by NM, 3-Aug-1999.)
 |-  ( ph  ->  ch )   &    |-  ( et  ->  th )   &    |-  ( ps  ->  ( ( ch  /\  th )  ->  ta ) )   =>    |-  ( ps  ->  ( ( ph  /\  et )  ->  ta ) )
 
Theoremsylan9 638 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 ) )
 
Theoremsylan9r 639 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ch  ->  ta ) )   =>    |-  ( ( th  /\  ph )  ->  ( ps  ->  ta ) )
 
Theoremmtand 640 A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ph  ->  -.  ps )
 
Theoremmtord 641 A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  -.  th )   &    |-  ( ph  ->  ( ps  ->  ( ch  \/  th ) ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremsyl2anc 642 Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylancl 643 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  (
 ( ps  /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylancr 644 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |- 
 ps   &    |-  ( ph  ->  ch )   &    |-  (
 ( ps  /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylanbrc 645 Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( th  <->  ( ps  /\  ch ) )   =>    |-  ( ph  ->  th )
 
Theoremsylancb 646 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
 |-  ( ph  <->  ps )   &    |-  ( ph  <->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylancbr 647 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
 |-  ( ps  <->  ph )   &    |-  ( ch  <->  ph )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremsylancom 648 Syllogism inference with commutation of antecedents. (Contributed by NM, 2-Jul-2008.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ch 
 /\  ps )  ->  th )   =>    |-  (
 ( ph  /\  ps )  ->  th )
 
Theoremmpdan 649 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 )
 
Theoremmpancom 650 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 )
 
Theoremmpan 651 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 )
 
Theoremmpan2 652 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 )
 
Theoremmp2an 653 An inference based on modus ponens. (Contributed by NM, 13-Apr-1995.)
 |-  ph   &    |- 
 ps   &    |-  ( ( ph  /\  ps )  ->  ch )   =>    |- 
 ch
 
Theoremmp4an 654 An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2010.)
 |-  ph   &    |- 
 ps   &    |- 
 ch   &    |- 
 th   &    |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |- 
 ta
 
Theoremmpan2d 655 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
 |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ( ps 
 /\  ch )  ->  th )
 )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremmpand 656 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 ) )
 
Theoremmpani 657 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 )
 )
 
Theoremmpan2i 658 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 )
 )
 
Theoremmp2ani 659 An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)
 |- 
 ps   &    |- 
 ch   &    |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  th )
 
Theoremmp2and 660 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  th )
 
Theoremmpanl1 661 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 )
 
Theoremmpanl2 662 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 )
 
Theoremmpanl12 663 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
 |-  ph   &    |- 
 ps   &    |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ch  ->  th )
 
Theoremmpanr1 664 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 )
 
Theoremmpanr2 665 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 )
 
Theoremmpanr12 666 An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.)
 |- 
 ps   &    |- 
 ch   &    |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ph  ->  th )
 
Theoremmpanlr1 667 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 )
 
Theorempm5.74da 668 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 4-May-2007.)
 |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ( ps  ->  ch )  <->  ( ps  ->  th )
 ) )
 
Theorempm4.45 669 Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  <->  ( ph  /\  ( ph  \/  ps ) ) )
 
Theoremimdistan 670 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 ) ) )
 
Theoremimdistani 671 Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( ph  /\  ch ) )
 
Theoremimdistanri 672 Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ps 
 /\  ph )  ->  ( ch  /\  ph ) )
 
Theoremimdistand 673 Distribution of implication with conjunction (deduction rule). (Contributed by NM, 27-Aug-2004.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  ->  ( ps  /\  th ) ) )
 
Theoremimdistanda 674 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 ) ) )
 
Theoremanbi2i 675 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 ) )
 
Theoremanbi1i 676 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 ) )
 
Theoremanbi2ci 677 Variant of anbi2i 675 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ph  /\  ch ) 
 <->  ( ch  /\  ps ) )
 
Theoremanbi12i 678 Conjoin both sides of two equivalences. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  /\  ch ) 
 <->  ( ps  /\  th ) )
 
Theoremanbi12ci 679 Variant of anbi12i 678 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  /\  ch ) 
 <->  ( th  /\  ps ) )
 
Theoremsylan9bb 680 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ch  <->  ta ) )   =>    |-  ( ( ph  /\ 
 th )  ->  ( ps 
 <->  ta ) )
 
Theoremsylan9bbr 681 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ch  <->  ta ) )   =>    |-  ( ( th  /\  ph )  ->  ( ps  <->  ta ) )
 
Theoremorbi2d 682 Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( th  \/  ps ) 
 <->  ( th  \/  ch ) ) )
 
Theoremorbi1d 683 Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( ps  \/  th ) 
 <->  ( ch  \/  th ) ) )
 
Theoremanbi2d 684 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 ) ) )
 
Theoremanbi1d 685 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 ) ) )
 
Theoremorbi1 686 Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ph  \/  ch )  <->  ( ps  \/  ch )
 ) )
 
Theoremanbi1 687 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 )
 ) )
 
Theoremanbi2 688 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 16-Nov-2013.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ch  /\  ph )  <->  ( ch  /\  ps )
 ) )
 
Theorembitr 689 Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  <->  ps )  /\  ( ps  <->  ch ) )  ->  ( ph  <->  ch ) )
 
Theoremorbi12d 690 Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( ps  \/  th ) 
 <->  ( ch  \/  ta ) ) )
 
Theoremanbi12d 691 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 ) ) )
 
Theorempm5.3 692 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 ) ) )
 
Theorempm5.61 693 Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.)
 |-  ( ( ( ph  \/  ps )  /\  -.  ps )  <->  ( ph  /\  -.  ps ) )
 
Theoremadantll 694 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 )
 
Theoremadantlr 695 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 )
 
Theoremadantrl 696 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 )
 
Theoremadantrr 697 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 )
 
Theoremadantlll 698 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 )
 
Theoremadantllr 699 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 )
 
Theoremadantlrl 700 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 )
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