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	simplld 528	Deduction form of simpll 527, eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	simplr 529	Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜓)
Theorem	simplrd 530	Deduction eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∧ 𝜃))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	simprl 531	Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜓)
Theorem	simprld 532	Deduction eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → (𝜓 ∧ (𝜒 ∧ 𝜃)))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	simprr 533	Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜒)
Theorem	simprrd 534	Deduction form of simprr 533, eliminating a double conjunct. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → (𝜓 ∧ (𝜒 ∧ 𝜃)))    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	simplll 535	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜑)
Theorem	simpllr 536	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜓)
Theorem	simplrl 537	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜓)
Theorem	simplrr 538	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜒)
Theorem	simprll 539	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ ((𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)) → 𝜓)
Theorem	simprlr 540	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ ((𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)) → 𝜒)
Theorem	simprrl 541	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ 𝜃))) → 𝜒)
Theorem	simprrr 542	Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
⊢ ((𝜑 ∧ (𝜓 ∧ (𝜒 ∧ 𝜃))) → 𝜃)
Theorem	simp-4l 543	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜑)
Theorem	simp-4r 544	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)
Theorem	simp-5l 545	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ ((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜑)
Theorem	simp-5r 546	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ ((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜓)
Theorem	simp-6l 547	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜑)
Theorem	simp-6r 548	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜓)
Theorem	simp-7l 549	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ ((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜑)
Theorem	simp-7r 550	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ ((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜓)
Theorem	simp-8l 551	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜑)
Theorem	simp-8r 552	Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
⊢ (((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) → 𝜓)
Theorem	simp-9l 553	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 554	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 555	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 556	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 557	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 558	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	pm4.87 559	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 560	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 561	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 562	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 563	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 564	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 565	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 566	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 567	Swap two conjuncts in antecedent. The label suffix "s" means that an12 563 is combined with syl 14 (or a variant). (Contributed by NM, 13-Mar-1996.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜒)) → 𝜃)
Theorem	ancom2s 568	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 569	Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)    ⇒   ⊢ ((𝜒 ∧ (𝜓 ∧ 𝜑)) → 𝜃)
Theorem	an32s 570	Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜓) → 𝜃)
Theorem	ancom1s 571	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 572	Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)    ⇒   ⊢ (((𝜒 ∧ 𝜓) ∧ 𝜑) → 𝜃)
Theorem	anass1rs 573	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 574	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 575	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 576	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 577	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 578	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 579	Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
⊢ (((𝜓 ∧ 𝜑) ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss5 580	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 581	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 582	Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.)
⊢ (𝜑 → ((𝜓 ∧ 𝜑) → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabsi7 583	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 584	Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.)
⊢ (𝜓 → ((𝜓 ∧ 𝜑) → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	anabss7 585	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 586	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 587	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 588	Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)))
Theorem	an42 589	Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)))
Theorem	an43 590	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 591	A rearrangement of conjuncts. (Contributed by Rodolfo Medina, 25-Sep-2010.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → (𝜑 ∧ 𝜃))
Theorem	an4s 592	Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) → 𝜏)
Theorem	an42s 593	Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)) → 𝜏)
Theorem	anandi 594	Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)))
Theorem	anandir 595	Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒)))
Theorem	anandis 596	Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜒)) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜏)
Theorem	anandirs 597	Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒)) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜏)
Theorem	syl2an2 598	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 599	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 600	Deduce an equivalence from two implications. (Contributed by NM, 17-Feb-2007.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))