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Theorem List for Metamath Proof Explorer - 601-700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnan 601 Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
((𝜑 → ¬ (𝜓𝜒)) ↔ ((𝜑𝜓) → ¬ 𝜒))
 
Theorempm4.15 602 Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
(((𝜑𝜓) → ¬ 𝜒) ↔ ((𝜓𝜒) → ¬ 𝜑))
 
Theorempm4.78 603 Implication distributes over disjunction. Theorem *4.78 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
(((𝜑𝜓) ∨ (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒)))
 
Theorempm4.79 604 Theorem *4.79 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2013.)
(((𝜓𝜑) ∨ (𝜒𝜑)) ↔ ((𝜓𝜒) → 𝜑))
 
Theorempm4.87 605 Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.)
(((((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒))) ∧ ((𝜑 → (𝜓𝜒)) ↔ (𝜓 → (𝜑𝜒)))) ∧ ((𝜓 → (𝜑𝜒)) ↔ ((𝜓𝜑) → 𝜒)))
 
Theorempm3.33 606 Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))
 
Theorempm3.34 607 Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((𝜓𝜒) ∧ (𝜑𝜓)) → (𝜑𝜒))
 
Theorempm3.35 608 Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. (Contributed by NM, 14-Dec-2002.)
((𝜑 ∧ (𝜑𝜓)) → 𝜓)
 
Theorempm5.31 609 Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜒 ∧ (𝜑𝜓)) → (𝜑 → (𝜓𝜒)))
 
Theoremimp4b 610 An importation inference. (Contributed by NM, 26-Apr-1994.) Shorten imp4a 611. (Revised by Wolf Lammen, 19-Jul-2021.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑𝜓) → ((𝜒𝜃) → 𝜏))
 
Theoremimp4a 611 An importation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Jul-2021.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (𝜓 → ((𝜒𝜃) → 𝜏)))
 
Theoremimp4aOLD 612 Obsolete proof of imp4a 611 as of 19-Jul-2021. (Contributed by NM, 26-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (𝜓 → ((𝜒𝜃) → 𝜏)))
 
Theoremimp4bOLD 613 Obsolete proof of imp4b 610 as of 19-Jul-2021. (Contributed by NM, 26-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑𝜓) → ((𝜒𝜃) → 𝜏))
 
Theoremimp4c 614 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → (((𝜓𝜒) ∧ 𝜃) → 𝜏))
 
Theoremimp4d 615 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → ((𝜓 ∧ (𝜒𝜃)) → 𝜏))
 
Theoremimp41 616 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theoremimp42 617 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)
 
Theoremimp43 618 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)
 
Theoremimp44 619 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜏)
 
Theoremimp45 620 An importation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜏)
 
Theoremimp5a 621 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜑 → (𝜓 → (𝜒 → ((𝜃𝜏) → 𝜂))))
 
Theoremimp5d 622 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (((𝜑𝜓) ∧ 𝜒) → ((𝜃𝜏) → 𝜂))
 
Theoremimp5g 623 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       ((𝜑𝜓) → (((𝜒𝜃) ∧ 𝜏) → 𝜂))
 
Theoremimp55 624 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) ∧ 𝜏) → 𝜂)
 
Theoremimp511 625 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       ((𝜑 ∧ ((𝜓 ∧ (𝜒𝜃)) ∧ 𝜏)) → 𝜂)
 
Theoremexpimpd 626 Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → 𝜃))
 
Theoremexp31 627 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremexp32 628 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremexp4b 629 An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) Shorten exp4a 630. (Revised by Wolf Lammen, 20-Jul-2021.)
((𝜑𝜓) → ((𝜒𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp4a 630 An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 20-Jul-2021.)
(𝜑 → (𝜓 → ((𝜒𝜃) → 𝜏)))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp4aOLD 631 Obsolete proof of exp4a 630 as of 20-Jul-2021. (Contributed by NM, 26-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → ((𝜒𝜃) → 𝜏)))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp4bOLD 632 Obsolete proof of exp4b 629 as of 20-Jul-2021. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp4c 633 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → (((𝜓𝜒) ∧ 𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp4d 634 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(𝜑 → ((𝜓 ∧ (𝜒𝜃)) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp41 635 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp42 636 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp43 637 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp44 638 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp45 639 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexpr 640 Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓) → (𝜒𝜃))
 
Theoremexp5c 641 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → ((𝜓𝜒) → ((𝜃𝜏) → 𝜂)))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp5j 642 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → ((((𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp5l 643 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (((𝜓𝜒) ∧ (𝜃𝜏)) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp53 644 An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)
((((𝜑𝜓) ∧ (𝜒𝜃)) ∧ 𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexpl 645 Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (𝜑 → ((𝜓𝜒) → 𝜃))
 
Theoremimpr 646 Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
 
Theoremimpl 647 Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.)
(𝜑 → ((𝜓𝜒) → 𝜃))       (((𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theoremimpac 648 Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → (𝜒𝜓))
 
Theoremexbiri 649 Inference form of exbir 37598. This proof is exbiriVD 38004 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → (𝜓 → (𝜃𝜒)))
 
Theoremsimprbda 650 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓) → 𝜒)
 
Theoremsimplbda 651 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓) → 𝜃)
 
Theoremsimplbi2 652 Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜒𝜑))
 
Theoremsimplbi2comt 653 Closed form of simplbi2com 654. (Contributed by Alan Sare, 22-Jul-2012.)
((𝜑 ↔ (𝜓𝜒)) → (𝜒 → (𝜓𝜑)))
 
Theoremsimplbi2com 654 A deduction eliminating a conjunct, similar to simplbi2 652. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)
(𝜑 ↔ (𝜓𝜒))       (𝜒 → (𝜓𝜑))
 
Theoremsimpl2im 655 Implication from an eliminated conjunct implied by the antecedent. (Contributed by BJ/AV, 5-Apr-2021.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑𝜃)
 
Theoremsimplbiim 656 Implication from an eliminated conjunct equivalent to the antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒))    &   (𝜒𝜃)       (𝜑𝜃)
 
Theoremdfbi2 657 A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49. (Contributed by NM, 24-Jan-1993.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
 
Theoremdfbi 658 Definition df-bi 195 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.)
(((𝜑𝜓) → ((𝜑𝜓) ∧ (𝜓𝜑))) ∧ (((𝜑𝜓) ∧ (𝜓𝜑)) → (𝜑𝜓)))
 
Theorempm4.71 659 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.)
((𝜑𝜓) ↔ (𝜑 ↔ (𝜑𝜓)))
 
Theorempm4.71r 660 Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 25-Jul-1999.)
((𝜑𝜓) ↔ (𝜑 ↔ (𝜓𝜑)))
 
Theorempm4.71i 661 Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 4-Jan-2004.)
(𝜑𝜓)       (𝜑 ↔ (𝜑𝜓))
 
Theorempm4.71ri 662 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.)
(𝜑𝜓)       (𝜑 ↔ (𝜓𝜑))
 
Theorempm4.71d 663 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.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 ↔ (𝜓𝜒)))
 
Theorempm4.71rd 664 Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 10-Feb-2005.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 ↔ (𝜒𝜓)))
 
Theorempm5.32 665 Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.)
((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
 
Theorempm5.32i 666 Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) ↔ (𝜑𝜒))
 
Theorempm5.32ri 667 Distribution of implication over biconditional (inference rule). (Contributed by NM, 12-Mar-1995.)
(𝜑 → (𝜓𝜒))       ((𝜓𝜑) ↔ (𝜒𝜑))
 
Theorempm5.32d 668 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 29-Oct-1996.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
 
Theorempm5.32rd 669 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 25-Dec-2004.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜓)))
 
Theorempm5.32da 670 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 9-Dec-2006.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
 
Theorembiadan2 671 Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.)
(𝜑𝜓)    &   (𝜓 → (𝜑𝜒))       (𝜑 ↔ (𝜓𝜒))
 
Theorempm4.24 672 Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.)
(𝜑 ↔ (𝜑𝜑))
 
Theoremanidm 673 Idempotent law for conjunction. (Contributed by NM, 8-Jan-2004.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)
((𝜑𝜑) ↔ 𝜑)
 
Theoremanidms 674 Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.)
((𝜑𝜑) → 𝜓)       (𝜑𝜓)
 
Theoremanidmdbi 675 Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)
((𝜑 → (𝜓𝜓)) ↔ (𝜑𝜓))
 
Theoremanasss 676 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
 
Theoremanassrs 677 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       (((𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theoremanass 678 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.)
(((𝜑𝜓) ∧ 𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
 
Theoremsylanl1 679 A syllogism inference. (Contributed by NM, 10-Mar-2005.)
(𝜑𝜓)    &   (((𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜒) ∧ 𝜃) → 𝜏)
 
Theoremsylanl2 680 A syllogism inference. (Contributed by NM, 1-Jan-2005.)
(𝜑𝜒)    &   (((𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜓𝜑) ∧ 𝜃) → 𝜏)
 
Theoremsylanr1 681 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
(𝜑𝜒)    &   ((𝜓 ∧ (𝜒𝜃)) → 𝜏)       ((𝜓 ∧ (𝜑𝜃)) → 𝜏)
 
Theoremsylanr2 682 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
(𝜑𝜃)    &   ((𝜓 ∧ (𝜒𝜃)) → 𝜏)       ((𝜓 ∧ (𝜒𝜑)) → 𝜏)
 
Theoremsylani 683 A syllogism inference. (Contributed by NM, 2-May-1996.)
(𝜑𝜒)    &   (𝜓 → ((𝜒𝜃) → 𝜏))       (𝜓 → ((𝜑𝜃) → 𝜏))
 
Theoremsylan2i 684 A syllogism inference. (Contributed by NM, 1-Aug-1994.)
(𝜑𝜃)    &   (𝜓 → ((𝜒𝜃) → 𝜏))       (𝜓 → ((𝜒𝜑) → 𝜏))
 
Theoremsyl2ani 685 A syllogism inference. (Contributed by NM, 3-Aug-1999.)
(𝜑𝜒)    &   (𝜂𝜃)    &   (𝜓 → ((𝜒𝜃) → 𝜏))       (𝜓 → ((𝜑𝜂) → 𝜏))
 
Theoremsylan9 686 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜑𝜃) → (𝜓𝜏))
 
Theoremsylan9r 687 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜃𝜑) → (𝜓𝜏))
 
Theoremmtand 688 A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.)
(𝜑 → ¬ 𝜒)    &   ((𝜑𝜓) → 𝜒)       (𝜑 → ¬ 𝜓)
 
Theoremmtord 689 A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → ¬ 𝜃)    &   (𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ¬ 𝜓)
 
Theoremsyl2anc 690 Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
TheoremhypstkdOLD 691 Obsolete proof of mpidan 700 as of 28-Mar-2021. (Contributed by Stanislas Polu, 9-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)    &   (𝜑𝜒)       ((𝜑𝜓) → 𝜃)
 
Theoremsylancl 692 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝜑𝜓)    &   𝜒    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylancr 693 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝜓    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylanbrc 694 Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜃 ↔ (𝜓𝜒))       (𝜑𝜃)
 
Theoremsylancb 695 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylancbr 696 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
(𝜓𝜑)    &   (𝜒𝜑)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylancom 697 Syllogism inference with commutation of antecedents. (Contributed by NM, 2-Jul-2008.)
((𝜑𝜓) → 𝜒)    &   ((𝜒𝜓) → 𝜃)       ((𝜑𝜓) → 𝜃)
 
Theoremmpdan 698 An inference based on modus ponens. (Contributed by NM, 23-May-1999.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
(𝜑𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)
 
Theoremmpancom 699 An inference based on modus ponens with commutation of antecedents. (Contributed by NM, 28-Oct-2003.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
(𝜓𝜑)    &   ((𝜑𝜓) → 𝜒)       (𝜓𝜒)
 
Theoremmpidan 700 A deduction which "stacks" a hypothesis. (Contributed by Stanislas Polu, 9-Mar-2020.) (Proof shortened by Wolf Lammen, 28-Mar-2021.)
(𝜑𝜒)    &   (((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)
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