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Theorem List for Metamath Proof Explorer - 501-600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsimplbi2 501 Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜒𝜑))
 
Theoremsimplbi2comt 502 Closed form of simplbi2com 503. (Contributed by Alan Sare, 22-Jul-2012.)
((𝜑 ↔ (𝜓𝜒)) → (𝜒 → (𝜓𝜑)))
 
Theoremsimplbi2com 503 A deduction eliminating a conjunct, similar to simplbi2 501. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)
(𝜑 ↔ (𝜓𝜒))       (𝜒 → (𝜓𝜑))
 
Theoremsimpl2im 504 Implication from an eliminated conjunct implied by the antecedent. (Contributed by BJ/AV, 5-Apr-2021.) (Proof shortened by Wolf Lammen, 26-Mar-2022.)
(𝜑 → (𝜓𝜒))    &   (𝜒𝜃)       (𝜑𝜃)
 
Theoremsimplbiim 505 Implication from an eliminated conjunct equivalent to the antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 26-Mar-2022.)
(𝜑 ↔ (𝜓𝜒))    &   (𝜒𝜃)       (𝜑𝜃)
 
Theoremimpel 506 An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen, 2-Sep-2020.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜓)       ((𝜑𝜃) → 𝜒)
 
Theoremmpan9 507 Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑𝜓)    &   (𝜒 → (𝜓𝜃))       ((𝜑𝜒) → 𝜃)
 
Theoremsylan9 508 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 509 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜃𝜑) → (𝜓𝜏))
 
Theoremsylan9bb 510 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜑𝜃) → (𝜓𝜏))
 
Theoremsylan9bbr 511 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))       ((𝜃𝜑) → (𝜓𝜏))
 
Theoremjca 512 Deduce conjunction of the consequents of two implications ("join consequents with 'and'"). Deduction form of pm3.2 470 and pm3.2i 471. Its associated deduction is jcad 513. Equivalent to the natural deduction rule I ( introduction), see natded 28110. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
(𝜑𝜓)    &   (𝜑𝜒)       (𝜑 → (𝜓𝜒))
 
Theoremjcad 513 Deduction conjoining the consequents of two implications. Deduction form of jca 512 and double deduction form of pm3.2 470 and pm3.2i 471. (Contributed by NM, 15-Jul-1993.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremjca2 514 Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(𝜑 → (𝜓𝜒))    &   (𝜓𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremjca31 515 Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)       (𝜑 → ((𝜓𝜒) ∧ 𝜃))
 
Theoremjca32 516 Join three consequents. (Contributed by FL, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)       (𝜑 → (𝜓 ∧ (𝜒𝜃)))
 
Theoremjcai 517 Deduction replacing implication with conjunction. (Contributed by NM, 15-Jul-1993.)
(𝜑𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓𝜒))
 
Theoremjcab 518 Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 27-Nov-2013.)
((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
 
Theorempm4.76 519 Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∧ (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒)))
 
Theoremjctil 520 Inference conjoining a theorem to left of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
(𝜑𝜓)    &   𝜒       (𝜑 → (𝜒𝜓))
 
Theoremjctir 521 Inference conjoining a theorem to right of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
(𝜑𝜓)    &   𝜒       (𝜑 → (𝜓𝜒))
 
Theoremjccir 522 Inference conjoining a consequent of a consequent to the right of the consequent in an implication. See also ex-natded5.3i 28116. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by AV, 20-Aug-2019.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑 → (𝜓𝜒))
 
Theoremjccil 523 Inference conjoining a consequent of a consequent to the left of the consequent in an implication. Remark: One can also prove this theorem using syl 17 and jca 512 (as done in jccir 522), which would be 4 bytes shorter, but one step longer than the current proof. (Proof modification is discouraged.) (Contributed by AV, 20-Aug-2019.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑 → (𝜒𝜓))
 
Theoremjctl 524 Inference conjoining a theorem to the left of a consequent. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
𝜓       (𝜑 → (𝜓𝜑))
 
Theoremjctr 525 Inference conjoining a theorem to the right of a consequent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
𝜓       (𝜑 → (𝜑𝜓))
 
Theoremjctild 526 Deduction conjoining a theorem to left of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)       (𝜑 → (𝜓 → (𝜃𝜒)))
 
Theoremjctird 527 Deduction conjoining a theorem to right of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremiba 528 Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.)
(𝜑 → (𝜓 ↔ (𝜓𝜑)))
 
Theoremibar 529 Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.)
(𝜑 → (𝜓 ↔ (𝜑𝜓)))
 
Theorembiantru 530 A wff is equivalent to its conjunction with truth. (Contributed by NM, 26-May-1993.)
𝜑       (𝜓 ↔ (𝜓𝜑))
 
Theorembiantrur 531 A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.)
𝜑       (𝜓 ↔ (𝜑𝜓))
 
Theorembiantrud 532 A wff is equivalent to its conjunction with truth. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.)
(𝜑𝜓)       (𝜑 → (𝜒 ↔ (𝜒𝜓)))
 
Theorembiantrurd 533 A wff is equivalent to its conjunction with truth. (Contributed by NM, 1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑𝜓)       (𝜑 → (𝜒 ↔ (𝜓𝜒)))
 
Theorembianfi 534 A wff conjoined with falsehood is false. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)
¬ 𝜑       (𝜑 ↔ (𝜓𝜑))
 
Theorembianfd 535 A wff conjoined with falsehood is false. (Contributed by NM, 27-Mar-1995.) (Proof shortened by Wolf Lammen, 5-Nov-2013.)
(𝜑 → ¬ 𝜓)       (𝜑 → (𝜓 ↔ (𝜓𝜒)))
 
Theorembaib 536 Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜑𝜒))
 
Theorembaibr 537 Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜒𝜑))
 
Theoremrbaibr 538 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)
(𝜑 ↔ (𝜓𝜒))       (𝜒 → (𝜓𝜑))
 
Theoremrbaib 539 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)
(𝜑 ↔ (𝜓𝜒))       (𝜒 → (𝜑𝜓))
 
Theorembaibd 540 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜒) → (𝜓𝜃))
 
Theoremrbaibd 541 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜃) → (𝜓𝜒))
 
Theorembianabs 542 Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.)
(𝜑 → (𝜓 ↔ (𝜑𝜒)))       (𝜑 → (𝜓𝜒))
 
Theorempm5.44 543 Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜑 → (𝜓𝜒))))
 
Theorempm5.42 544 Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (𝜓 → (𝜑𝜒))))
 
Theoremancl 545 Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)
((𝜑𝜓) → (𝜑 → (𝜑𝜓)))
 
Theoremanclb 546 Conjoin antecedent to left of consequent. Theorem *4.7 of [WhiteheadRussell] p. 120. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
((𝜑𝜓) ↔ (𝜑 → (𝜑𝜓)))
 
Theoremancr 547 Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
((𝜑𝜓) → (𝜑 → (𝜓𝜑)))
 
Theoremancrb 548 Conjoin antecedent to right of consequent. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
((𝜑𝜓) ↔ (𝜑 → (𝜓𝜑)))
 
Theoremancli 549 Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.)
(𝜑𝜓)       (𝜑 → (𝜑𝜓))
 
Theoremancri 550 Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
(𝜑𝜓)       (𝜑 → (𝜓𝜑))
 
Theoremancld 551 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜓𝜒)))
 
Theoremancrd 552 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜒𝜓)))
 
Theoremimpac 553 Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → (𝜒𝜓))
 
Theoremanc2l 554 Conjoin antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 14-Jul-2013.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜑𝜒))))
 
Theoremanc2r 555 Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜒𝜑))))
 
Theoremanc2li 556 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜑𝜒)))
 
Theoremanc2ri 557 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜒𝜑)))
 
Theorempm4.71 558 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 559 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 560 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 561 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 562 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 563 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)))
 
Theorempm4.24 564 Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.)
(𝜑 ↔ (𝜑𝜑))
 
Theoremanidm 565 Idempotent law for conjunction. (Contributed by NM, 8-Jan-2004.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)
((𝜑𝜑) ↔ 𝜑)
 
Theoremanidmdbi 566 Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)
((𝜑 → (𝜓𝜓)) ↔ (𝜑𝜓))
 
Theoremanidms 567 Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.)
((𝜑𝜑) → 𝜓)       (𝜑𝜓)
 
Theoremimdistan 568 Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.)
((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))
 
Theoremimdistani 569 Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → (𝜑𝜒))
 
Theoremimdistanri 570 Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)
(𝜑 → (𝜓𝜒))       ((𝜓𝜑) → (𝜒𝜑))
 
Theoremimdistand 571 Distribution of implication with conjunction (deduction form). (Contributed by NM, 27-Aug-2004.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))
 
Theoremimdistanda 572 Distribution of implication with conjunction (deduction version with conjoined antecedent). (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))
 
Theorempm5.3 573 Theorem *5.3 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(((𝜑𝜓) → 𝜒) ↔ ((𝜑𝜓) → (𝜑𝜒)))
 
Theorempm5.32 574 Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.)
((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
 
Theorempm5.32i 575 Distribution of implication over biconditional (inference form). (Contributed by NM, 1-Aug-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) ↔ (𝜑𝜒))
 
Theorempm5.32ri 576 Distribution of implication over biconditional (inference form). (Contributed by NM, 12-Mar-1995.)
(𝜑 → (𝜓𝜒))       ((𝜓𝜑) ↔ (𝜒𝜑))
 
Theorempm5.32d 577 Distribution of implication over biconditional (deduction form). (Contributed by NM, 29-Oct-1996.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
 
Theorempm5.32rd 578 Distribution of implication over biconditional (deduction form). (Contributed by NM, 25-Dec-2004.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜓)))
 
Theorempm5.32da 579 Distribution of implication over biconditional (deduction form). (Contributed by NM, 9-Dec-2006.)
((𝜑𝜓) → (𝜒𝜃))       (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
 
Theoremsylan 580 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
(𝜑𝜓)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)
 
Theoremsylanb 581 A syllogism inference. (Contributed by NM, 18-May-1994.)
(𝜑𝜓)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)
 
Theoremsylanbr 582 A syllogism inference. (Contributed by NM, 18-May-1994.)
(𝜓𝜑)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜒) → 𝜃)
 
Theoremsylanbrc 583 Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜃 ↔ (𝜓𝜒))       (𝜑𝜃)
 
Theoremsyl2anc 584 Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsyl2anc2 585 Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023.)
(𝜑𝜓)    &   (𝜓𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylancl 586 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝜑𝜓)    &   𝜒    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylancr 587 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝜓    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylancom 588 Syllogism inference with commutation of antecedents. (Contributed by NM, 2-Jul-2008.)
((𝜑𝜓) → 𝜒)    &   ((𝜒𝜓) → 𝜃)       ((𝜑𝜓) → 𝜃)
 
Theoremsylanblc 589 Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
(𝜑𝜓)    &   𝜒    &   ((𝜓𝜒) ↔ 𝜃)       (𝜑𝜃)
 
Theoremsylanblrc 590 Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
(𝜑𝜓)    &   𝜒    &   (𝜃 ↔ (𝜓𝜒))       (𝜑𝜃)
 
Theoremsyldan 591 A syllogism deduction with conjoined antecedents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜒) → 𝜃)       ((𝜑𝜓) → 𝜃)
 
Theoremsylan2 592 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
(𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜓𝜑) → 𝜃)
 
Theoremsylan2b 593 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
(𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜓𝜑) → 𝜃)
 
Theoremsylan2br 594 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
(𝜒𝜑)    &   ((𝜓𝜒) → 𝜃)       ((𝜓𝜑) → 𝜃)
 
Theoremsyl2an 595 A double syllogism inference. For an implication-only version, see syl2im 40. (Contributed by NM, 31-Jan-1997.)
(𝜑𝜓)    &   (𝜏𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜏) → 𝜃)
 
Theoremsyl2anr 596 A double syllogism inference. For an implication-only version, see syl2imc 41. (Contributed by NM, 17-Sep-2013.)
(𝜑𝜓)    &   (𝜏𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜏𝜑) → 𝜃)
 
Theoremsyl2anb 597 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
(𝜑𝜓)    &   (𝜏𝜒)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜏) → 𝜃)
 
Theoremsyl2anbr 598 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
(𝜓𝜑)    &   (𝜒𝜏)    &   ((𝜓𝜒) → 𝜃)       ((𝜑𝜏) → 𝜃)
 
Theoremsylancb 599 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theoremsylancbr 600 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
(𝜓𝜑)    &   (𝜒𝜑)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
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