P. 7 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 601-700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sylbida 601	A syllogism deduction. (Contributed by SN, 16-Jul-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	sylan2 602	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 603	A syllogism inference. (Contributed by NM, 21-Apr-1994.)
⊢ (𝜑 ↔ 𝜒)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → 𝜃)
Theorem	sylan2br 604	A syllogism inference. (Contributed by NM, 21-Apr-1994.)
⊢ (𝜒 ↔ 𝜑)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → 𝜃)
Theorem	syl2an 605	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 606	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 607	A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜏 ↔ 𝜒)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜏) → 𝜃)
Theorem	syl2anbr 608	A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜒 ↔ 𝜏)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜏) → 𝜃)
Theorem	sylancb 609	A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜑 ↔ 𝜒)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	sylancbr 610	A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜒 ↔ 𝜑)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	syldanl 611	A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜏)
Theorem	syland 612	A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜏))
Theorem	sylani 613	A syllogism inference. (Contributed by NM, 2-May-1996.)
⊢ (𝜑 → 𝜒)    &   ⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))    ⇒   ⊢ (𝜓 → ((𝜑 ∧ 𝜃) → 𝜏))
Theorem	sylan2d 614	A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → ((𝜃 ∧ 𝜒) → 𝜏))    ⇒   ⊢ (𝜑 → ((𝜃 ∧ 𝜓) → 𝜏))
Theorem	sylan2i 615	A syllogism inference. (Contributed by NM, 1-Aug-1994.)
⊢ (𝜑 → 𝜃)    &   ⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))    ⇒   ⊢ (𝜓 → ((𝜒 ∧ 𝜑) → 𝜏))
Theorem	syl2ani 616	A syllogism inference. (Contributed by NM, 3-Aug-1999.)
⊢ (𝜑 → 𝜒)    &   ⊢ (𝜂 → 𝜃)    &   ⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))    ⇒   ⊢ (𝜓 → ((𝜑 ∧ 𝜂) → 𝜏))
Theorem	syl2and 617	A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜃 → 𝜏))    &   ⊢ (𝜑 → ((𝜒 ∧ 𝜏) → 𝜂))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜂))
Theorem	anim12d 618	Conjoin antecedents and consequents in a deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 18-Dec-2013.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜃 → 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜒 ∧ 𝜏)))
Theorem	anim12d1 619	Variant of anim12d 618 where the second implication does not depend on the antecedent. (Contributed by Rodolfo Medina, 12-Oct-2010.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → 𝜏)    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜒 ∧ 𝜏)))
Theorem	anim1d 620	Add a conjunct to right of antecedent and consequent in a deduction. (Contributed by NM, 3-Apr-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜒 ∧ 𝜃)))
Theorem	anim2d 621	Add a conjunct to left of antecedent and consequent in a deduction. (Contributed by NM, 14-May-1993.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ((𝜃 ∧ 𝜓) → (𝜃 ∧ 𝜒)))
Theorem	anim12i 622	Conjoin antecedents and consequents of two premises. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 14-Dec-2013.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃))
Theorem	anim12ci 623	Variant of anim12i 622 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → (𝜃 ∧ 𝜓))
Theorem	anim1i 624	Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒))
Theorem	anim1ci 625	Introduce conjunct to both sides of an implication. (Contributed by Peter Mazsa, 24-Sep-2022.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → (𝜒 ∧ 𝜓))
Theorem	anim2i 626	Introduce conjunct to both sides of an implication. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜓))
Theorem	anim12ii 627	Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → (𝜓 → 𝜏))    ⇒   ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → (𝜒 ∧ 𝜏)))
Theorem	anim12dan 628	Conjoin antecedents and consequents in a deduction. (Contributed by Jeff Madsen, 16-Jun-2011.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜃)) → (𝜒 ∧ 𝜏))
Theorem	im2anan9 629	Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) → (𝜒 ∧ 𝜂)))
Theorem	im2anan9r 630	Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 → (𝜏 → 𝜂))    ⇒   ⊢ ((𝜃 ∧ 𝜑) → ((𝜓 ∧ 𝜏) → (𝜒 ∧ 𝜂)))
Theorem	pm3.45 631	Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → 𝜓) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒)))
Theorem	anbi2i 632	Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜒 ∧ 𝜑) ↔ (𝜒 ∧ 𝜓))
Theorem	anbi1i 633	Introduce a right conjunct to both sides of a logical equivalence. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))
Theorem	anbi2ci 634	Variant of anbi2i 632 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓))
Theorem	anbi1ci 635	Variant of anbi1i 633 with commutation. (Contributed by Peter Mazsa, 7-Mar-2020.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜒 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒))
Theorem	bianbi 636	Exchanging conjunction in a biconditional. (Contributed by Peter Mazsa, 31-Jul-2023.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (𝜑 ↔ (𝜃 ∧ 𝜒))
Theorem	anbi12i 637	Conjoin both sides of two equivalences. (Contributed by NM, 12-Mar-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃))
Theorem	anbi12ci 638	Variant of anbi12i 637 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜃 ∧ 𝜓))
Theorem	anbi2d 639	Deduction adding a left conjunct to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜃 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒)))
Theorem	anbi1d 640	Deduction adding a right conjunct to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜃) ↔ (𝜒 ∧ 𝜃)))
Theorem	anbi12d 641	Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜃) ↔ (𝜒 ∧ 𝜏)))
Theorem	anbi1 642	Introduce a right conjunct to both sides of a logical equivalence. Theorem *4.36 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒)))
Theorem	anbi2 643	Introduce a left conjunct to both sides of a logical equivalence. (Contributed by NM, 16-Nov-2013.)
⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ∧ 𝜑) ↔ (𝜒 ∧ 𝜓)))
Theorem	anbi1cd 644	Introduce a proposition as left conjunct on the left-hand side and right conjunct on the right-hand side of an equivalence. Deduction form. (Contributed by Peter Mazsa, 22-May-2021.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ((𝜃 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))
Theorem	an2anr 645	Double commutation in conjunction. (Contributed by Peter Mazsa, 27-Jun-2019.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜓 ∧ 𝜑) ∧ (𝜃 ∧ 𝜒)))
Theorem	pm4.38 646	Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → ((𝜑 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))
Theorem	bi2anan9 647	Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 31-Jul-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 → (𝜏 ↔ 𝜂))    ⇒   ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂)))
Theorem	bi2anan9r 648	Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 → (𝜏 ↔ 𝜂))    ⇒   ⊢ ((𝜃 ∧ 𝜑) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂)))
Theorem	bi2bian9 649	Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 → (𝜏 ↔ 𝜂))    ⇒   ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ↔ 𝜏) ↔ (𝜒 ↔ 𝜂)))
Theorem	anbiim 650	Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.) (Proof shortened by Wolf Lammen, 7-May-2025.) (Proof shortened by Garrett Katz, 15-Jun-2026.)
⊢ (𝜑 → (𝜒 → 𝜃))    &   ⊢ (𝜓 → (𝜃 → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
Theorem	anbiimOLD 651	Obsolete version of anbiim 650 as of 15-Jun-2026. Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.) (Proof shortened by Wolf Lammen, 7-May-2025.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝜑 → (𝜒 → 𝜃))    &   ⊢ (𝜓 → (𝜃 → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
Theorem	bianass 652	An inference to merge two lists of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ ((𝜂 ∧ 𝜑) ↔ ((𝜂 ∧ 𝜓) ∧ 𝜒))
Theorem	bianassc 653	An inference to merge two lists of conjuncts. (Contributed by Peter Mazsa, 24-Sep-2022.)
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))    ⇒   ⊢ ((𝜂 ∧ 𝜑) ↔ ((𝜓 ∧ 𝜂) ∧ 𝜒))
Theorem	an21 654	Swap two conjuncts. (Contributed by Peter Mazsa, 18-Sep-2022.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
Theorem	an12 655	Swap two conjuncts. Note that the first digit (1) in the label refers to the outer conjunct position, and the next digit (2) to the inner conjunct position. (Contributed by NM, 12-Mar-1995.) (Proof shortened by Peter Mazsa, 18-Sep-2022.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
Theorem	an32 656	A rearrangement of conjuncts. (Contributed by NM, 12-Mar-1995.) (Proof shortened by Wolf Lammen, 25-Dec-2012.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ 𝜓))
Theorem	an13 657	A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜒 ∧ (𝜓 ∧ 𝜑)))
Theorem	an31 658	A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜒 ∧ 𝜓) ∧ 𝜑))
Theorem	an12s 659	Swap two conjuncts in antecedent. The label suffix "s" means that an12 655 is combined with syl 17 (or a variant). (Contributed by NM, 13-Mar-1996.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜒)) → 𝜃)
Theorem	ancom2s 660	Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜓)) → 𝜃)
Theorem	an13s 661	Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜑)) → 𝜃)
Theorem	an32s 662	Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜓) → 𝜃)
Theorem	ancom1s 663	Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜒) → 𝜃)
Theorem	an31s 664	Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜒 ∧ 𝜓) ∧ 𝜑) → 𝜃)
Theorem	anass1rs 665	Commutative-associative law for conjunction in an antecedent. (Contributed by Jeff Madsen, 19-Jun-2011.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜓) → 𝜃)
Theorem	an4 666	Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)))
Theorem	an42 667	Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)))
Theorem	an43 668	Rearrangement of 4 conjuncts. (Contributed by Rodolfo Medina, 24-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜒)))
Theorem	an3 669	A rearrangement of conjuncts. (Contributed by Rodolfo Medina, 25-Sep-2010.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → (𝜑 ∧ 𝜃))
Theorem	an4s 670	Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) → 𝜏)
Theorem	an42s 671	Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)) → 𝜏)
Theorem	anabs1 672	Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜑) ↔ (𝜑 ∧ 𝜓))
Theorem	anabs5 673	Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓))
Theorem	anabs7 674	Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.)
⊢ ((𝜓 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓))
Theorem	anabsan 675	Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.)
⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss1 676	Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜑) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss4 677	Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
⊢ (((𝜓 ∧ 𝜑) ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss5 678	Absorption of antecedent into conjunction. (Contributed by NM, 10-May-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabsi5 679	Absorption of antecedent into conjunction. (Contributed by NM, 11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
⊢ (𝜑 → ((𝜑 ∧ 𝜓) → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabsi6 680	Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.)
⊢ (𝜑 → ((𝜓 ∧ 𝜑) → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabsi7 681	Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
⊢ (𝜓 → ((𝜑 ∧ 𝜓) → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabsi8 682	Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.)
⊢ (𝜓 → ((𝜓 ∧ 𝜑) → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss7 683	Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 19-Nov-2013.)
⊢ ((𝜓 ∧ (𝜑 ∧ 𝜓)) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabsan2 684	Absorption of antecedent with conjunction. (Contributed by NM, 10-May-2004.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜓)) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss3 685	Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anandi 686	Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))
Theorem	anandir 687	Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒)))
Theorem	anandis 688	Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜏)
Theorem	anandirs 689	Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜏)
Theorem	sylanl1 690	A syllogism inference. (Contributed by NM, 10-Mar-2005.)
⊢ (𝜑 → 𝜓)    &   ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)
Theorem	sylanl2 691	A syllogism inference. (Contributed by NM, 1-Jan-2005.)
⊢ (𝜑 → 𝜒)    &   ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜃) → 𝜏)
Theorem	sylanr1 692	A syllogism inference. (Contributed by NM, 9-Apr-2005.)
⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → 𝜏)
Theorem	sylanr2 693	A syllogism inference. (Contributed by NM, 9-Apr-2005.)
⊢ (𝜑 → 𝜃)    &   ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜑)) → 𝜏)
Theorem	syl6an 694	A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜒 → 𝜃))    &   ⊢ ((𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ (𝜑 → (𝜒 → 𝜏))
Theorem	syl2an2r 695	syl2anr 606 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.) (Proof shortened by Wolf Lammen, 28-Mar-2022.)
⊢ (𝜑 → 𝜓)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    &   ⊢ ((𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → 𝜏)
Theorem	syl2an2 696	syl2an 605 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
⊢ (𝜑 → 𝜓)    &   ⊢ ((𝜒 ∧ 𝜑) → 𝜃)    &   ⊢ ((𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜒 ∧ 𝜑) → 𝜏)
Theorem	mpdan 697	An inference based on modus ponens. (Contributed by NM, 23-May-1999.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
⊢ (𝜑 → 𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	mpancom 698	An inference based on modus ponens with commutation of antecedents. (Contributed by NM, 28-Oct-2003.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
⊢ (𝜓 → 𝜑)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜓 → 𝜒)
Theorem	mpidan 699	A deduction which "stacks" a hypothesis. (Contributed by Stanislas Polu, 9-Mar-2020.) (Proof shortened by Wolf Lammen, 28-Mar-2021.)
⊢ (𝜑 → 𝜒)    &   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
Theorem	mpan 700	An inference based on modus ponens. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
⊢ 𝜑    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜓 → 𝜒)