P. 6 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 501-600   *Has distinct variable group(s)

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	birani 506	Inference adding a conjunct to the left-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → 𝜓)
Theorem	bilani 507	Inference adding a conjunct to the left-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜒 ∧ 𝜑) → 𝜓)
Theorem	biranri 508	Inference adding a conjunct to the right-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜓 ∧ 𝜒) → 𝜑)
Theorem	bilanri 509	Inference adding a conjunct to the right-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜒 ∧ 𝜓) → 𝜑)
Theorem	simpl2im 510	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 511	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 512	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 513	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 514	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 515	Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → (𝜒 → 𝜏))    ⇒   ⊢ ((𝜃 ∧ 𝜑) → (𝜓 → 𝜏))
Theorem	sylan9bb 516	Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 → (𝜒 ↔ 𝜏))    ⇒   ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜏))
Theorem	sylan9bbr 517	Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 → (𝜒 ↔ 𝜏))    ⇒   ⊢ ((𝜃 ∧ 𝜑) → (𝜓 ↔ 𝜏))
Theorem	jca 518	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 519. Equivalent to the natural deduction rule ∧ I (∧ introduction), see natded 30544. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ∧ 𝜒))
Theorem	jcad 519	Deduction conjoining the consequents of two implications. Deduction form of jca 518 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 520	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 521	Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∧ 𝜃))
Theorem	jca32 522	Join three consequents. (Contributed by FL, 1-Aug-2009.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 ∧ (𝜒 ∧ 𝜃)))
Theorem	jcai 523	Deduction replacing implication with conjunction. (Contributed by NM, 15-Jul-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ∧ 𝜒))
Theorem	jcab 524	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	pm4.76 525	Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒)))
Theorem	jctil 526	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 527	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 528	Inference conjoining a consequent of a consequent to the right of the consequent in an implication. See also ex-natded5.3i 30550. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by AV, 20-Aug-2019.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ∧ 𝜒))
Theorem	jccil 529	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 518 (as done in jccir 528), 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 530	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 531	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 532	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 533	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 534	Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.)
⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑)))
Theorem	ibar 535	Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.)
⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
Theorem	biantru 536	A wff is equivalent to its conjunction with truth. (Contributed by NM, 26-May-1993.)
⊢ 𝜑    ⇒   ⊢ (𝜓 ↔ (𝜓 ∧ 𝜑))
Theorem	biantrur 537	A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.)
⊢ 𝜑    ⇒   ⊢ (𝜓 ↔ (𝜑 ∧ 𝜓))
Theorem	biantrud 538	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 539	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 540	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 541	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 542	Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜓 → (𝜑 ↔ 𝜒))
Theorem	baibr 543	Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜓 → (𝜒 ↔ 𝜑))
Theorem	rbaibr 544	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 545	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 546	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃))
Theorem	rbaibd 547	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜒))
Theorem	bianabs 548	Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.)
⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜒)))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	pm5.44 549	Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → 𝜓) → ((𝜑 → 𝜒) ↔ (𝜑 → (𝜓 ∧ 𝜒))))
Theorem	pm5.42 550	Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜑 → (𝜓 → (𝜑 ∧ 𝜒))))
Theorem	ancl 551	Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜑 ∧ 𝜓)))
Theorem	anclb 552	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 553	Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜓 ∧ 𝜑)))
Theorem	ancrb 554	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 555	Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜑 ∧ 𝜓))
Theorem	ancri 556	Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ∧ 𝜑))
Theorem	ancld 557	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 558	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 559	Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜓))
Theorem	anc2l 560	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 561	Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (𝜓 → (𝜒 ∧ 𝜑))))
Theorem	anc2li 562	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 563	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	pm4.71 564	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 565	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 566	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 567	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 568	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 569	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	pm4.24 570	Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.)
⊢ (𝜑 ↔ (𝜑 ∧ 𝜑))
Theorem	anidm 571	Idempotent law for conjunction. (Contributed by NM, 8-Jan-2004.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)
⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑)
Theorem	anidmdbi 572	Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)
⊢ ((𝜑 → (𝜓 ∧ 𝜓)) ↔ (𝜑 → 𝜓))
Theorem	anidms 573	Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.)
⊢ ((𝜑 ∧ 𝜑) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	imdistan 574	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 575	Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒))
Theorem	imdistanri 576	Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ ((𝜓 ∧ 𝜑) → (𝜒 ∧ 𝜑))
Theorem	imdistand 577	Distribution of implication with conjunction (deduction form). (Contributed by NM, 27-Aug-2004.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜃)))
Theorem	imdistanda 578	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	pm5.3 579	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	pm5.32 580	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 581	Distribution of implication over biconditional (inference form). (Contributed by NM, 1-Aug-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))
Theorem	pm5.32ri 582	Distribution of implication over biconditional (inference form). (Contributed by NM, 12-Mar-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜒 ∧ 𝜑))
Theorem	bianim 583	Exchanging conjunction in a biconditional. (Contributed by Peter Mazsa, 31-Jul-2023.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 ↔ (𝜃 ∧ 𝜒))
Theorem	pm5.32d 584	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 585	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 586	Distribution of implication over biconditional (deduction form). (Contributed by NM, 9-Dec-2006.)
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃)))
Theorem	bian1d 587	Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017.) (Proof shortened by Hongxiu Chen, 29-Jun-2025.) (Proof shortened by Peter Mazsa, 24-Feb-2026.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))
Theorem	sylan 588	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 589	A syllogism inference. (Contributed by NM, 18-May-1994.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
Theorem	sylanbr 590	A syllogism inference. (Contributed by NM, 18-May-1994.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
Theorem	sylanbrc 591	Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜃 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	syl2anc 592	Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	syl2anc2 593	Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 → 𝜒)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	sylancl 594	Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝜑 → 𝜓)    &   ⊢ 𝜒    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	sylancr 595	Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝜓    &   ⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	sylancom 596	Syllogism inference with commutation of antecedents. (Contributed by NM, 2-Jul-2008.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜒 ∧ 𝜓) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	sylanblc 597	Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
⊢ (𝜑 → 𝜓)    &   ⊢ 𝜒    &   ⊢ ((𝜓 ∧ 𝜒) ↔ 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	sylanblrc 598	Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
⊢ (𝜑 → 𝜓)    &   ⊢ 𝜒    &   ⊢ (𝜃 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	syldan 599	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	sylbida 600	A syllogism deduction. (Contributed by SN, 16-Jul-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)