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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | simprbda 501 | Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.) |
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
Theorem | simplbda 502 | Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.) |
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜃) | ||
Theorem | simplbi2 503 | Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜓 → (𝜒 → 𝜑)) | ||
Theorem | simplbi2comt 504 | Closed form of simplbi2com 505. (Contributed by Alan Sare, 22-Jul-2012.) |
⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑))) | ||
Theorem | simplbi2com 505 | A deduction eliminating a conjunct, similar to simplbi2 503. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜒 → (𝜓 → 𝜑)) | ||
Theorem | simpl2im 506 | 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 → (𝜓 ∧ 𝜒)) & ⊢ (𝜒 → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | simplbiim 507 | 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 ↔ (𝜓 ∧ 𝜒)) & ⊢ (𝜒 → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | impel 508 | An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen, 2-Sep-2020.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜃 → 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝜃) → 𝜒) | ||
Theorem | mpan9 509 | Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → (𝜓 → 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜒) → 𝜃) | ||
Theorem | sylan9 510 | Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜃 → (𝜒 → 𝜏)) ⇒ ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → 𝜏)) | ||
Theorem | sylan9r 511 | Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜃 → (𝜒 → 𝜏)) ⇒ ⊢ ((𝜃 ∧ 𝜑) → (𝜓 → 𝜏)) | ||
Theorem | sylan9bb 512 | Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜃 → (𝜒 ↔ 𝜏)) ⇒ ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜏)) | ||
Theorem | sylan9bbr 513 | Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜃 → (𝜒 ↔ 𝜏)) ⇒ ⊢ ((𝜃 ∧ 𝜑) → (𝜓 ↔ 𝜏)) | ||
Theorem | jca 514 | Deduce conjunction of the consequents of two implications ("join consequents with 'and'"). Deduction form of pm3.2 472 and pm3.2i 473. Its associated deduction is jcad 515. Equivalent to the natural deduction rule ∧ I (∧ introduction), see natded 28184. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) ⇒ ⊢ (𝜑 → (𝜓 ∧ 𝜒)) | ||
Theorem | jcad 515 | Deduction conjoining the consequents of two implications. Deduction form of jca 514 and double deduction form of pm3.2 472 and pm3.2i 473. (Contributed by NM, 15-Jul-1993.) (Proof shortened by Wolf Lammen, 23-Jul-2013.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜓 → 𝜃)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃))) | ||
Theorem | jca2 516 | Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜓 → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃))) | ||
Theorem | jca31 517 | Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∧ 𝜃)) | ||
Theorem | jca32 518 | Join three consequents. (Contributed by FL, 1-Aug-2009.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 ∧ (𝜒 ∧ 𝜃))) | ||
Theorem | jcai 519 | Deduction replacing implication with conjunction. (Contributed by NM, 15-Jul-1993.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 ∧ 𝜒)) | ||
Theorem | jcab 520 | 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 → (𝜓 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒))) | ||
Theorem | pm4.76 521 | Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒))) | ||
Theorem | jctil 522 | Inference conjoining a theorem to left of consequent in an implication. (Contributed by NM, 31-Dec-1993.) |
⊢ (𝜑 → 𝜓) & ⊢ 𝜒 ⇒ ⊢ (𝜑 → (𝜒 ∧ 𝜓)) | ||
Theorem | jctir 523 | Inference conjoining a theorem to right of consequent in an implication. (Contributed by NM, 31-Dec-1993.) |
⊢ (𝜑 → 𝜓) & ⊢ 𝜒 ⇒ ⊢ (𝜑 → (𝜓 ∧ 𝜒)) | ||
Theorem | jccir 524 | Inference conjoining a consequent of a consequent to the right of the consequent in an implication. See also ex-natded5.3i 28190. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by AV, 20-Aug-2019.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ (𝜑 → (𝜓 ∧ 𝜒)) | ||
Theorem | jccil 525 | 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 514 (as done in jccir 524), which would be 4 bytes shorter, but one step longer than the current proof. (Proof modification is discouraged.) (Contributed by AV, 20-Aug-2019.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ (𝜑 → (𝜒 ∧ 𝜓)) | ||
Theorem | jctl 526 | 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 → (𝜓 ∧ 𝜑)) | ||
Theorem | jctr 527 | 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 → (𝜑 ∧ 𝜓)) | ||
Theorem | jctild 528 | Deduction conjoining a theorem to left of consequent in an implication. (Contributed by NM, 21-Apr-2005.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 ∧ 𝜒))) | ||
Theorem | jctird 529 | Deduction conjoining a theorem to right of consequent in an implication. (Contributed by NM, 21-Apr-2005.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃))) | ||
Theorem | iba 530 | Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.) |
⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑))) | ||
Theorem | ibar 531 | Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.) |
⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓))) | ||
Theorem | biantru 532 | A wff is equivalent to its conjunction with truth. (Contributed by NM, 26-May-1993.) |
⊢ 𝜑 ⇒ ⊢ (𝜓 ↔ (𝜓 ∧ 𝜑)) | ||
Theorem | biantrur 533 | A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.) |
⊢ 𝜑 ⇒ ⊢ (𝜓 ↔ (𝜑 ∧ 𝜓)) | ||
Theorem | biantrud 534 | A wff is equivalent to its conjunction with truth. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝜒 ↔ (𝜒 ∧ 𝜓))) | ||
Theorem | biantrurd 535 | A wff is equivalent to its conjunction with truth. (Contributed by NM, 1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝜒 ↔ (𝜓 ∧ 𝜒))) | ||
Theorem | bianfi 536 | A wff conjoined with falsehood is false. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 26-Nov-2012.) |
⊢ ¬ 𝜑 ⇒ ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑)) | ||
Theorem | bianfd 537 | 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 ↔ (𝜓 ∧ 𝜒))) | ||
Theorem | baib 538 | Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜓 → (𝜑 ↔ 𝜒)) | ||
Theorem | baibr 539 | Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜓 → (𝜒 ↔ 𝜑)) | ||
Theorem | rbaibr 540 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) (Proof shortened by Wolf Lammen, 19-Jan-2020.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜒 → (𝜓 ↔ 𝜑)) | ||
Theorem | rbaib 541 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) (Proof shortened by Wolf Lammen, 19-Jan-2020.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜒 → (𝜑 ↔ 𝜓)) | ||
Theorem | baibd 542 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃)) | ||
Theorem | rbaibd 543 | Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) ⇒ ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜒)) | ||
Theorem | bianabs 544 | Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.) |
⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜒))) ⇒ ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | ||
Theorem | pm5.44 545 | Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → 𝜓) → ((𝜑 → 𝜒) ↔ (𝜑 → (𝜓 ∧ 𝜒)))) | ||
Theorem | pm5.42 546 | Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜑 → (𝜓 → (𝜑 ∧ 𝜒)))) | ||
Theorem | ancl 547 | Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.) |
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜑 ∧ 𝜓))) | ||
Theorem | anclb 548 | 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.) |
⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜑 ∧ 𝜓))) | ||
Theorem | ancr 549 | Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.) |
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜓 ∧ 𝜑))) | ||
Theorem | ancrb 550 | Conjoin antecedent to right of consequent. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.) |
⊢ ((𝜑 → 𝜓) ↔ (𝜑 → (𝜓 ∧ 𝜑))) | ||
Theorem | ancli 551 | Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝜑 ∧ 𝜓)) | ||
Theorem | ancri 552 | Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝜓 ∧ 𝜑)) | ||
Theorem | ancld 553 | 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 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜓 ∧ 𝜒))) | ||
Theorem | ancrd 554 | 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 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜓))) | ||
Theorem | impac 555 | Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜓)) | ||
Theorem | anc2l 556 | Conjoin antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 14-Jul-2013.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (𝜓 → (𝜑 ∧ 𝜒)))) | ||
Theorem | anc2r 557 | Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (𝜓 → (𝜒 ∧ 𝜑)))) | ||
Theorem | anc2li 558 | 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 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜒))) | ||
Theorem | anc2ri 559 | 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 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜑))) | ||
Theorem | pm4.71 560 | 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.) |
⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓))) | ||
Theorem | pm4.71r 561 | Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 25-Jul-1999.) |
⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜓 ∧ 𝜑))) | ||
Theorem | pm4.71i 562 | Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 4-Jan-2004.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 ↔ (𝜑 ∧ 𝜓)) | ||
Theorem | pm4.71ri 563 | 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.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑)) | ||
Theorem | pm4.71d 564 | 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 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒))) | ||
Theorem | pm4.71rd 565 | 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 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜓))) | ||
Theorem | pm4.24 566 | Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.) |
⊢ (𝜑 ↔ (𝜑 ∧ 𝜑)) | ||
Theorem | anidm 567 | Idempotent law for conjunction. (Contributed by NM, 8-Jan-2004.) (Proof shortened by Wolf Lammen, 14-Mar-2014.) |
⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑) | ||
Theorem | anidmdbi 568 | Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.) |
⊢ ((𝜑 → (𝜓 ∧ 𝜓)) ↔ (𝜑 → 𝜓)) | ||
Theorem | anidms 569 | Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.) |
⊢ ((𝜑 ∧ 𝜑) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | imdistan 570 | Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒))) | ||
Theorem | imdistani 571 | Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒)) | ||
Theorem | imdistanri 572 | Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ ((𝜓 ∧ 𝜑) → (𝜒 ∧ 𝜑)) | ||
Theorem | imdistand 573 | Distribution of implication with conjunction (deduction form). (Contributed by NM, 27-Aug-2004.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜃))) | ||
Theorem | imdistanda 574 | Distribution of implication with conjunction (deduction version with conjoined antecedent). (Contributed by Jeff Madsen, 19-Jun-2011.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜃))) | ||
Theorem | pm5.3 575 | Theorem *5.3 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒))) | ||
Theorem | pm5.32 576 | Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) |
⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) | ||
Theorem | pm5.32i 577 | Distribution of implication over biconditional (inference form). (Contributed by NM, 1-Aug-1994.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)) | ||
Theorem | pm5.32ri 578 | Distribution of implication over biconditional (inference form). (Contributed by NM, 12-Mar-1995.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜒 ∧ 𝜑)) | ||
Theorem | pm5.32d 579 | Distribution of implication over biconditional (deduction form). (Contributed by NM, 29-Oct-1996.) |
⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃))) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃))) | ||
Theorem | pm5.32rd 580 | Distribution of implication over biconditional (deduction form). (Contributed by NM, 25-Dec-2004.) |
⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃))) ⇒ ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜓))) | ||
Theorem | pm5.32da 581 | Distribution of implication over biconditional (deduction form). (Contributed by NM, 9-Dec-2006.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃))) | ||
Theorem | sylan 582 | A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.) |
⊢ (𝜑 → 𝜓) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜒) → 𝜃) | ||
Theorem | sylanb 583 | A syllogism inference. (Contributed by NM, 18-May-1994.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜒) → 𝜃) | ||
Theorem | sylanbr 584 | A syllogism inference. (Contributed by NM, 18-May-1994.) |
⊢ (𝜓 ↔ 𝜑) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜒) → 𝜃) | ||
Theorem | sylanbrc 585 | Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜃 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | syl2anc 586 | Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | syl2anc2 587 | Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜓 → 𝜒) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | sylancl 588 | Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝜑 → 𝜓) & ⊢ 𝜒 & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | sylancr 589 | Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ 𝜓 & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | sylancom 590 | Syllogism inference with commutation of antecedents. (Contributed by NM, 2-Jul-2008.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜒 ∧ 𝜓) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜃) | ||
Theorem | sylanblc 591 | Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.) |
⊢ (𝜑 → 𝜓) & ⊢ 𝜒 & ⊢ ((𝜓 ∧ 𝜒) ↔ 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | sylanblrc 592 | Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.) |
⊢ (𝜑 → 𝜓) & ⊢ 𝜒 & ⊢ (𝜃 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | syldan 593 | A syllogism deduction with conjoined antecedents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜃) | ||
Theorem | sylan2 594 | A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.) |
⊢ (𝜑 → 𝜒) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝜃) | ||
Theorem | sylan2b 595 | A syllogism inference. (Contributed by NM, 21-Apr-1994.) |
⊢ (𝜑 ↔ 𝜒) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝜃) | ||
Theorem | sylan2br 596 | A syllogism inference. (Contributed by NM, 21-Apr-1994.) |
⊢ (𝜒 ↔ 𝜑) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝜃) | ||
Theorem | syl2an 597 | A double syllogism inference. For an implication-only version, see syl2im 40. (Contributed by NM, 31-Jan-1997.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜏 → 𝜒) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜏) → 𝜃) | ||
Theorem | syl2anr 598 | A double syllogism inference. For an implication-only version, see syl2imc 41. (Contributed by NM, 17-Sep-2013.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜏 → 𝜒) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜏 ∧ 𝜑) → 𝜃) | ||
Theorem | syl2anb 599 | A double syllogism inference. (Contributed by NM, 29-Jul-1999.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜏 ↔ 𝜒) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜏) → 𝜃) | ||
Theorem | syl2anbr 600 | A double syllogism inference. (Contributed by NM, 29-Jul-1999.) |
⊢ (𝜓 ↔ 𝜑) & ⊢ (𝜒 ↔ 𝜏) & ⊢ ((𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜏) → 𝜃) |
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