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	sylancbr 601	A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜒 ↔ 𝜑)    &   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	syldanl 602	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 603	A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜏))
Theorem	sylani 604	A syllogism inference. (Contributed by NM, 2-May-1996.)
⊢ (𝜑 → 𝜒)    &   ⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))    ⇒   ⊢ (𝜓 → ((𝜑 ∧ 𝜃) → 𝜏))
Theorem	sylan2d 605	A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → ((𝜃 ∧ 𝜒) → 𝜏))    ⇒   ⊢ (𝜑 → ((𝜃 ∧ 𝜓) → 𝜏))
Theorem	sylan2i 606	A syllogism inference. (Contributed by NM, 1-Aug-1994.)
⊢ (𝜑 → 𝜃)    &   ⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))    ⇒   ⊢ (𝜓 → ((𝜒 ∧ 𝜑) → 𝜏))
Theorem	syl2ani 607	A syllogism inference. (Contributed by NM, 3-Aug-1999.)
⊢ (𝜑 → 𝜒)    &   ⊢ (𝜂 → 𝜃)    &   ⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))    ⇒   ⊢ (𝜓 → ((𝜑 ∧ 𝜂) → 𝜏))
Theorem	syl2and 608	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 609	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 610	Variant of anim12d 609 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 611	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 612	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 613	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 614	Variant of anim12i 613 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → (𝜃 ∧ 𝜓))
Theorem	anim1i 615	Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒))
Theorem	anim1ci 616	Introduce conjunct to both sides of an implication. (Contributed by Peter Mazsa, 24-Sep-2022.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → (𝜒 ∧ 𝜓))
Theorem	anim2i 617	Introduce conjunct to both sides of an implication. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜓))
Theorem	anim12ii 618	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 619	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 620	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 621	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	pm3.45 622	Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → 𝜓) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒)))
Theorem	anbi2i 623	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 624	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 625	Variant of anbi2i 623 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 626	Variant of anbi1i 624 with commutation. (Contributed by Peter Mazsa, 7-Mar-2020.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜒 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒))
Theorem	anbi12i 627	Conjoin both sides of two equivalences. (Contributed by NM, 12-Mar-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃))
Theorem	anbi12ci 628	Variant of anbi12i 627 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜃 ∧ 𝜓))
Theorem	anbi2d 629	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 630	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 631	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 632	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 633	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 634	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 635	Double commutation in conjunction. (Contributed by Peter Mazsa, 27-Jun-2019.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜓 ∧ 𝜑) ∧ (𝜃 ∧ 𝜒)))
Theorem	pm4.38 636	Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
⊢ (((𝜑 ↔ 𝜒) ∧ (𝜓 ↔ 𝜃)) → ((𝜑 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))
Theorem	bi2anan9 637	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 638	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 639	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	bianass 640	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 641	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 642	Swap two conjuncts. (Contributed by Peter Mazsa, 18-Sep-2022.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
Theorem	an12 643	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 644	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 645	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 646	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 647	Swap two conjuncts in antecedent. The label suffix "s" means that an12 643 is combined with syl 17 (or a variant). (Contributed by NM, 13-Mar-1996.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜒)) → 𝜃)
Theorem	ancom2s 648	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 649	Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜑)) → 𝜃)
Theorem	an32s 650	Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜓) → 𝜃)
Theorem	ancom1s 651	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 652	Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜒 ∧ 𝜓) ∧ 𝜑) → 𝜃)
Theorem	anass1rs 653	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 654	Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)))
Theorem	an42 655	Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)))
Theorem	an43 656	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 657	A rearrangement of conjuncts. (Contributed by Rodolfo Medina, 25-Sep-2010.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → (𝜑 ∧ 𝜃))
Theorem	an4s 658	Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) → 𝜏)
Theorem	an42s 659	Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)) → 𝜏)
Theorem	anabs1 660	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 661	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 662	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 663	Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.)
⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss1 664	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 665	Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
⊢ (((𝜓 ∧ 𝜑) ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss5 666	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 667	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 668	Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.)
⊢ (𝜑 → ((𝜓 ∧ 𝜑) → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabsi7 669	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 670	Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.)
⊢ (𝜓 → ((𝜓 ∧ 𝜑) → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss7 671	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 672	Absorption of antecedent with conjunction. (Contributed by NM, 10-May-2004.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜓)) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss3 673	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 674	Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))
Theorem	anandir 675	Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒)))
Theorem	anandis 676	Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜏)
Theorem	anandirs 677	Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜏)
Theorem	sylanl1 678	A syllogism inference. (Contributed by NM, 10-Mar-2005.)
⊢ (𝜑 → 𝜓)    &   ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)
Theorem	sylanl2 679	A syllogism inference. (Contributed by NM, 1-Jan-2005.)
⊢ (𝜑 → 𝜒)    &   ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜃) → 𝜏)
Theorem	sylanr1 680	A syllogism inference. (Contributed by NM, 9-Apr-2005.)
⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → 𝜏)
Theorem	sylanr2 681	A syllogism inference. (Contributed by NM, 9-Apr-2005.)
⊢ (𝜑 → 𝜃)    &   ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜑)) → 𝜏)
Theorem	syl6an 682	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 683	syl2anr 597 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 684	syl2an 596 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 685	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 686	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 687	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 688	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)
Theorem	mpan2 689	An inference based on modus ponens. (Contributed by NM, 16-Sep-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
⊢ 𝜓    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	mp2an 690	An inference based on modus ponens. (Contributed by NM, 13-Apr-1995.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ 𝜒
Theorem	mp4an 691	An inference based on modus ponens. (Contributed by Jeff Madsen, 15-Jun-2010.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ 𝜃    &   ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ 𝜏
Theorem	mpan2d 692	A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	mpand 693	A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜃))
Theorem	mpani 694	An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
⊢ 𝜓    &   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜃))
Theorem	mpan2i 695	An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
⊢ 𝜒    &   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	mp2ani 696	An inference based on modus ponens. (Contributed by NM, 12-Dec-2004.)
⊢ 𝜓    &   ⊢ 𝜒    &   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	mp2and 697	A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	mpanl1 698	An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
⊢ 𝜑    &   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
Theorem	mpanl2 699	An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)
⊢ 𝜓    &   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
Theorem	mpanl12 700	An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
⊢ 𝜑    &   ⊢ 𝜓    &   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜒 → 𝜃)