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

Theoremexp5c 601 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp53 602 An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremexpl 603 Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.)

Theoremimpr 604 Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremimpl 605 Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremimpac 606 Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.)

Theoremexbiri 607 Inference form of exbir 1375. This proof is exbiriVD 29040 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.)

Theoremsimprbda 608 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)

Theoremsimplbda 609 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)

Theoremsimplbi2 610 Deduction eliminating a conjunct. Automatically derived from simplbi2VD 29032. (Contributed by Alan Sare, 31-Dec-2011.)

Theoremdfbi2 611 A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49. (Contributed by NM, 5-Aug-1993.)

Theoremdfbi 612 Definition df-bi 179 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.)

Theorempm4.71 613 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.)

Theorempm4.71r 614 Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 25-Jul-1999.)

Theorempm4.71i 615 Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 4-Jan-2004.)

Theorempm4.71ri 616 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.)

Theorempm4.71d 617 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.)

Theorempm4.71rd 618 Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 10-Feb-2005.)

Theorempm5.32 619 Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.)

Theorempm5.32i 620 Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)

Theorempm5.32ri 621 Distribution of implication over biconditional (inference rule). (Contributed by NM, 12-Mar-1995.)

Theorempm5.32d 622 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 29-Oct-1996.)

Theorempm5.32rd 623 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 25-Dec-2004.)

Theorempm5.32da 624 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 9-Dec-2006.)

Theorempm4.24 626 Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)

Theoremanidm 627 Idempotent law for conjunction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)

Theoremanidms 628 Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.)

Theoremanidmdbi 629 Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)

Theoremanasss 630 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)

Theoremanassrs 631 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)

Theoremanass 632 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.)

Theoremsylanl1 633 A syllogism inference. (Contributed by NM, 10-Mar-2005.)

Theoremsylanl2 634 A syllogism inference. (Contributed by NM, 1-Jan-2005.)

Theoremsylanr1 635 A syllogism inference. (Contributed by NM, 9-Apr-2005.)

Theoremsylanr2 636 A syllogism inference. (Contributed by NM, 9-Apr-2005.)

Theoremsylani 637 A syllogism inference. (Contributed by NM, 2-May-1996.)

Theoremsylan2i 638 A syllogism inference. (Contributed by NM, 1-Aug-1994.)

Theoremsyl2ani 639 A syllogism inference. (Contributed by NM, 3-Aug-1999.)

Theoremsylan9 640 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theoremsylan9r 641 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.)

Theoremmtand 642 A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)

Theoremmtord 643 A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.)

Theoremsyl2anc 644 Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.)

Theoremsylancl 645 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsylancr 646 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsylanbrc 647 Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremsylancb 648 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)

Theoremsylancbr 649 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)

Theoremsylancom 650 Syllogism inference with commutation of antecedents. (Contributed by NM, 2-Jul-2008.)

Theoremmpdan 651 An inference based on modus ponens. (Contributed by NM, 23-May-1999.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)

Theoremmpancom 652 An inference based on modus ponens with commutation of antecedents. (Contributed by NM, 28-Oct-2003.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Theoremmpan 653 An inference based on modus ponens. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Theoremmpan2 654 An inference based on modus ponens. (Contributed by NM, 16-Sep-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)

Theoremmp2an 655 An inference based on modus ponens. (Contributed by NM, 13-Apr-1995.)

Theoremmp4an 656 An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2010.)

Theoremmpan2d 657 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)

Theoremmpand 658 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Theoremmpani 659 An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)

Theoremmpan2i 660 An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)

Theoremmp2ani 661 An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)

Theoremmp2and 662 A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)

Theoremmpanl1 663 An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Theoremmpanl2 664 An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theoremmpanl12 665 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)

Theoremmpanr1 666 An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theoremmpanr2 667 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.)

Theoremmpanr12 668 An inference based on modus ponens. (Contributed by NM, 24-Jul-2009.)

Theoremmpanlr1 669 An inference based on modus ponens. (Contributed by NM, 30-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Theorempm5.74da 670 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 4-May-2007.)

Theorempm4.45 671 Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)

Theoremimdistan 672 Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.)

Theoremimdistani 673 Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)

Theoremimdistanri 674 Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)

Theoremimdistand 675 Distribution of implication with conjunction (deduction rule). (Contributed by NM, 27-Aug-2004.)

Theoremimdistanda 676 Distribution of implication with conjunction (deduction version with conjoined antecedent). (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremanbi2i 677 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.)

Theoremanbi1i 678 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.)

Theoremanbi2ci 679 Variant of anbi2i 677 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremanbi12i 680 Conjoin both sides of two equivalences. (Contributed by NM, 5-Aug-1993.)

Theoremanbi12ci 681 Variant of anbi12i 680 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsylan9bb 682 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)

Theoremsylan9bbr 683 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)

Theoremorbi2d 684 Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theoremorbi1d 685 Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theoremanbi2d 686 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.)

Theoremanbi1d 687 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.)

Theoremorbi1 688 Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)

Theoremanbi1 689 Introduce a right conjunct to both sides of a logical equivalence. Theorem *4.36 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)

Theoremanbi2 690 Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 16-Nov-2013.)

Theorembitr 691 Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)

Theoremorbi12d 692 Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 5-Aug-1993.)

Theoremanbi12d 693 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 5-Aug-1993.)

Theorempm5.3 694 Theorem *5.3 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theorempm5.61 695 Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.)

Theoremadantll 696 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)

Theoremadantlr 697 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)

Theoremadantrl 698 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)

Theoremadantrr 699 Deduction adding a conjunct to antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)

Theoremadantlll 700 Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 2-Dec-2012.)

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