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	simplbi 501	Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	simprbi 502	Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	simprbda 503	Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	simplbda 504	Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	simplbi2 505	Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜓 → (𝜒 → 𝜑))
Theorem	simplbi2comt 506	Closed form of simplbi2com 507. (Contributed by Alan Sare, 22-Jul-2012.)
⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑)))
Theorem	simplbi2com 507	A deduction eliminating a conjunct, similar to simplbi2 505. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜒 → (𝜓 → 𝜑))
Theorem	birani 508	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 509	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 510	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 511	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 512	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 513	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 514	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 515	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 516	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 517	Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 14-May-1993.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → (𝜒 → 𝜏))    ⇒   ⊢ ((𝜃 ∧ 𝜑) → (𝜓 → 𝜏))
Theorem	sylan9bb 518	Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 → (𝜒 ↔ 𝜏))    ⇒   ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜏))
Theorem	sylan9bbr 519	Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 → (𝜒 ↔ 𝜏))    ⇒   ⊢ ((𝜃 ∧ 𝜑) → (𝜓 ↔ 𝜏))
Theorem	jca 520	Deduce conjunction of the consequents of two implications ("join consequents with 'and'"). Deduction form of pm3.2 474 and pm3.2i 475. Its associated deduction is jcad 521. Equivalent to the natural deduction rule ∧ I (∧ introduction), see natded 30663. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ∧ 𝜒))
Theorem	jcad 521	Deduction conjoining the consequents of two implications. Deduction form of jca 520 and double deduction form of pm3.2 474 and pm3.2i 475. (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 522	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 523	Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∧ 𝜃))
Theorem	jca32 524	Join three consequents. (Contributed by FL, 1-Aug-2009.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 ∧ (𝜒 ∧ 𝜃)))
Theorem	jcai 525	Deduction replacing implication with conjunction. (Contributed by NM, 15-Jul-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ∧ 𝜒))
Theorem	jcab 526	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	pm4.76 527	Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒)))
Theorem	jctil 528	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 529	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 530	Inference conjoining a consequent of a consequent to the right of the consequent in an implication. See also ex-natded5.3i 30669. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by AV, 20-Aug-2019.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ∧ 𝜒))
Theorem	jccil 531	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 18 and jca 520 (as done in jccir 530), 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 532	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 533	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 534	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 535	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 536	Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.)
⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑)))
Theorem	ibar 537	Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.)
⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓)))
Theorem	biantru 538	A wff is equivalent to its conjunction with truth. (Contributed by NM, 26-May-1993.)
⊢ 𝜑    ⇒   ⊢ (𝜓 ↔ (𝜓 ∧ 𝜑))
Theorem	biantrur 539	A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.)
⊢ 𝜑    ⇒   ⊢ (𝜓 ↔ (𝜑 ∧ 𝜓))
Theorem	biantrud 540	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 541	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 542	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 543	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 544	Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜓 → (𝜑 ↔ 𝜒))
Theorem	baibr 545	Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜓 → (𝜒 ↔ 𝜑))
Theorem	rbaibr 546	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 547	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 548	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃))
Theorem	rbaibd 549	Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜒))
Theorem	bianabs 550	Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.)
⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜒)))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	pm5.44 551	Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → 𝜓) → ((𝜑 → 𝜒) ↔ (𝜑 → (𝜓 ∧ 𝜒))))
Theorem	pm5.42 552	Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜑 → (𝜓 → (𝜑 ∧ 𝜒))))
Theorem	ancl 553	Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜑 ∧ 𝜓)))
Theorem	anclb 554	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 555	Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜓 ∧ 𝜑)))
Theorem	ancrb 556	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 557	Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜑 ∧ 𝜓))
Theorem	ancri 558	Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ∧ 𝜑))
Theorem	ancld 559	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 560	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 561	Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜓))
Theorem	anc2l 562	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 563	Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (𝜓 → (𝜒 ∧ 𝜑))))
Theorem	anc2li 564	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 565	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	pm4.71 566	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 567	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 568	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 569	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 570	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 571	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	pm4.71da 572	Deduction converting a biconditional to a biconditional with conjunction. Variant of pm4.71d 570. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒)))
Theorem	pm4.24 573	Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.)
⊢ (𝜑 ↔ (𝜑 ∧ 𝜑))
Theorem	anidm 574	Idempotent law for conjunction. (Contributed by NM, 8-Jan-2004.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)
⊢ ((𝜑 ∧ 𝜑) ↔ 𝜑)
Theorem	anidmdbi 575	Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)
⊢ ((𝜑 → (𝜓 ∧ 𝜓)) ↔ (𝜑 → 𝜓))
Theorem	anidms 576	Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.)
⊢ ((𝜑 ∧ 𝜑) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	imdistan 577	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 578	Distribution of implication with conjunction. (Contributed by NM, 1-Aug-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜒))
Theorem	imdistanri 579	Distribution of implication with conjunction. (Contributed by NM, 8-Jan-2002.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ ((𝜓 ∧ 𝜑) → (𝜒 ∧ 𝜑))
Theorem	imdistand 580	Distribution of implication with conjunction (deduction form). (Contributed by NM, 27-Aug-2004.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜃)))
Theorem	imdistanda 581	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	pm5.3 582	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	pm5.32 583	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 584	Distribution of implication over biconditional (inference form). (Contributed by NM, 1-Aug-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))
Theorem	pm5.32ri 585	Distribution of implication over biconditional (inference form). (Contributed by NM, 12-Mar-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜒 ∧ 𝜑))
Theorem	bianim 586	Exchanging conjunction in a biconditional. (Contributed by Peter Mazsa, 31-Jul-2023.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 ↔ (𝜃 ∧ 𝜒))
Theorem	pm5.32d 587	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 588	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 589	Distribution of implication over biconditional (deduction form). (Contributed by NM, 9-Dec-2006.)
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃)))
Theorem	bian1d 590	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 591	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 592	A syllogism inference. (Contributed by NM, 18-May-1994.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
Theorem	sylanbr 593	A syllogism inference. (Contributed by NM, 18-May-1994.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
Theorem	sylanbrc 594	Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜃 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	syl2anc 595	Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	syl2anc2 596	Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 → 𝜒)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	sylancl 597	Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝜑 → 𝜓)    &   ⊢ 𝜒    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	sylancr 598	Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝜓    &   ⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	sylancom 599	Syllogism inference with commutation of antecedents. (Contributed by NM, 2-Jul-2008.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜒 ∧ 𝜓) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	sylanblc 600	Syllogism inference combined with a biconditional. (Contributed by BJ, 25-Apr-2019.)
⊢ (𝜑 → 𝜓)    &   ⊢ 𝜒    &   ⊢ ((𝜓 ∧ 𝜒) ↔ 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)