P. 6 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 501-600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ad7antlr 501	Deduction adding 7 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((((((((𝜒 ∧ 𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)
Theorem	ad8antr 502	Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (((((((((𝜑 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)
Theorem	ad8antlr 503	Deduction adding 8 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (((((((((𝜒 ∧ 𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)
Theorem	ad9antr 504	Deduction adding 9 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((((((((((𝜑 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜓)
Theorem	ad9antlr 505	Deduction adding 9 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((((((((((𝜒 ∧ 𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜓)
Theorem	ad10antr 506	Deduction adding 10 conjuncts to antecedent. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (((((((((((𝜑 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜓)
Theorem	ad10antlr 507	Deduction adding 10 conjuncts to antecedent. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (((((((((((𝜒 ∧ 𝜑) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜓)
Theorem	ad2ant2l 508	Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (((𝜃 ∧ 𝜑) ∧ (𝜏 ∧ 𝜓)) → 𝜒)
Theorem	ad2ant2r 509	Deduction adding two conjuncts to antecedent. (Contributed by NM, 8-Jan-2006.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (((𝜑 ∧ 𝜃) ∧ (𝜓 ∧ 𝜏)) → 𝜒)
Theorem	ad2ant2lr 510	Deduction adding two conjuncts to antecedent. (Contributed by NM, 23-Nov-2007.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (((𝜃 ∧ 𝜑) ∧ (𝜓 ∧ 𝜏)) → 𝜒)
Theorem	ad2ant2rl 511	Deduction adding two conjuncts to antecedent. (Contributed by NM, 24-Nov-2007.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (((𝜑 ∧ 𝜃) ∧ (𝜏 ∧ 𝜓)) → 𝜒)
Theorem	adantl3r 512	Deduction adding 1 conjunct to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((((𝜑 ∧ 𝜂) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
Theorem	ad4ant13 513	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((((𝜑 ∧ 𝜃) ∧ 𝜓) ∧ 𝜏) → 𝜒)
Theorem	ad4ant14 514	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((((𝜑 ∧ 𝜃) ∧ 𝜏) ∧ 𝜓) → 𝜒)
Theorem	ad4ant23 515	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((((𝜃 ∧ 𝜑) ∧ 𝜓) ∧ 𝜏) → 𝜒)
Theorem	ad4ant24 516	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((((𝜃 ∧ 𝜑) ∧ 𝜏) ∧ 𝜓) → 𝜒)
Theorem	adantl4r 517	Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ (((((𝜑 ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)    ⇒   ⊢ ((((((𝜑 ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)
Theorem	ad5ant12 518	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜒)
Theorem	ad5ant13 519	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (((((𝜑 ∧ 𝜃) ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) → 𝜒)
Theorem	ad5ant14 520	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (((((𝜑 ∧ 𝜃) ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) → 𝜒)
Theorem	ad5ant15 521	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (((((𝜑 ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜒)
Theorem	ad5ant23 522	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (((((𝜃 ∧ 𝜑) ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) → 𝜒)
Theorem	ad5ant24 523	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (((((𝜃 ∧ 𝜑) ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) → 𝜒)
Theorem	ad5ant25 524	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (((((𝜃 ∧ 𝜑) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜒)
Theorem	adantl5r 525	Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ ((((((𝜑 ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)    ⇒   ⊢ (((((((𝜑 ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)
Theorem	adantl6r 526	Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ (((((((𝜑 ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)    ⇒   ⊢ ((((((((𝜑 ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜅)
Theorem	simpll 527	Simplification of a conjunction. (Contributed by NM, 18-Mar-2007.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜑)
Theorem	simplr 528	Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜓)
Theorem	simprl 529	Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜓)
Theorem	simprld 530	Deduction eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → (𝜓 ∧ (𝜒 ∧ 𝜃)))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	simprr 531	Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜒)
Theorem	simprrd 532	Deduction form of simprr 531, eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → (𝜓 ∧ (𝜒 ∧ 𝜃)))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	simplll 533	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜑)
Theorem	simpllr 534	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜓)
Theorem	simplrl 535	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜓)
Theorem	simplrr 536	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜒)
Theorem	simprll 537	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ ((𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)) → 𝜓)
Theorem	simprlr 538	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ ((𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)) → 𝜒)
Theorem	simprrl 539	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ 𝜃))) → 𝜒)
Theorem	simprrr 540	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ 𝜃))) → 𝜃)
Theorem	simp-4l 541	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜑)
Theorem	simp-4r 542	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)
Theorem	simp-5l 543	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ ((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜑)
Theorem	simp-5r 544	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ ((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)
Theorem	simp-6l 545	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜑)
Theorem	simp-6r 546	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)
Theorem	simp-7l 547	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ ((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜑)
Theorem	simp-7r 548	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ ((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)
Theorem	simp-8l 549	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜑)
Theorem	simp-8r 550	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)
Theorem	simp-9l 551	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ ((((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜑)
Theorem	simp-9r 552	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ ((((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) → 𝜓)
Theorem	simp-10l 553	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (((((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜑)
Theorem	simp-10r 554	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (((((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) → 𝜓)
Theorem	simp-11l 555	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ ((((((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜑)
Theorem	simp-11r 556	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ ((((((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) → 𝜓)
Theorem	pm4.87 557	Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.)
⊢ (((((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒))) ∧ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜓 → (𝜑 → 𝜒)))) ∧ ((𝜓 → (𝜑 → 𝜒)) ↔ ((𝜓 ∧ 𝜑) → 𝜒)))
Theorem	a2and 558	Deduction distributing a conjunction as embedded antecedent. (Contributed by AV, 25-Oct-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)
⊢ (𝜑 → ((𝜓 ∧ 𝜌) → (𝜏 → 𝜃)))    &   ⊢ (𝜑 → ((𝜓 ∧ 𝜌) → 𝜒))    ⇒   ⊢ (𝜑 → (((𝜓 ∧ 𝜒) → 𝜏) → ((𝜓 ∧ 𝜌) → 𝜃)))
Theorem	animpimp2impd 559	Deduction deriving nested implications from conjunctions. (Contributed by AV, 21-Aug-2022.)
⊢ ((𝜓 ∧ 𝜑) → (𝜒 → (𝜃 → 𝜂)))    &   ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → (𝜂 → 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → (𝜃 → 𝜏))))
Theorem	abai 560	Introduce one conjunct as an antecedent to the other. "abai" stands for "and, biconditional, and, implication". (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝜑 → 𝜓)))
Theorem	an12 561	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.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
Theorem	an32 562	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 563	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 564	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 565	Swap two conjuncts in antecedent. The label suffix "s" means that an12 561 is combined with syl 14 (or a variant). (Contributed by NM, 13-Mar-1996.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜒)) → 𝜃)
Theorem	ancom2s 566	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 567	Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜑)) → 𝜃)
Theorem	an32s 568	Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜓) → 𝜃)
Theorem	ancom1s 569	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 570	Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜒 ∧ 𝜓) ∧ 𝜑) → 𝜃)
Theorem	anass1rs 571	Commutative-associative law for conjunction in an antecedent. (Contributed by Jeff Madsen, 19-Jun-2011.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜓) → 𝜃)
Theorem	anabs1 572	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 573	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 574	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 575	Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) (Revised by NM, 18-Nov-2013.)
⊢ (((𝜑 ∧ 𝜑) ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss1 576	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 577	Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
⊢ (((𝜓 ∧ 𝜑) ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss5 578	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 579	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 580	Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.)
⊢ (𝜑 → ((𝜓 ∧ 𝜑) → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabsi7 581	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 582	Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.)
⊢ (𝜓 → ((𝜓 ∧ 𝜑) → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss7 583	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 584	Absorption of antecedent with conjunction. (Contributed by NM, 10-May-2004.) (Revised by NM, 1-Jan-2013.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜓)) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss3 585	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	an4 586	Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)))
Theorem	an42 587	Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)))
Theorem	an4s 588	Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) → 𝜏)
Theorem	an42s 589	Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)) → 𝜏)
Theorem	anandi 590	Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))
Theorem	anandir 591	Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒)))
Theorem	anandis 592	Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜏)
Theorem	anandirs 593	Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜏)
Theorem	syl2an2 594	syl2an 289 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	syl2an2r 595	syl2anr 290 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
⊢ (𝜑 → 𝜓)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    &   ⊢ ((𝜓 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → 𝜏)
Theorem	impbida 596	Deduce an equivalence from two implications. (Contributed by NM, 17-Feb-2007.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	pm3.45 597	Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
⊢ ((𝜑 → 𝜓) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒)))
Theorem	im2anan9 598	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 599	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	anim12dan 600	Conjoin antecedents and consequents in a deduction. (Contributed by Mario Carneiro, 12-May-2014.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜃)) → (𝜒 ∧ 𝜏))