Theorem 13an22anass 1379

Description: Associative law for four conjunctions with a triple conjunction. (Contributed by Thierry Arnoux, 21-Jan-2025.)

Assertion

Ref	Expression
13an22anass	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))

Proof of Theorem 13an22anass

Step	Hyp	Ref	Expression
1		an2anr 648	. . 3 ⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜑)) ↔ ((𝜒 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)))
2		an2anr 648	. . . 4 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)) ↔ ((𝜒 ∧ 𝜑) ∧ (𝜓 ∧ 𝜃)))
3		an4 669	. . . 4 ⊢ (((𝜒 ∧ 𝜑) ∧ (𝜓 ∧ 𝜃)) ↔ ((𝜒 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)))
4	2, 3	bitri 278	. . 3 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)) ↔ ((𝜒 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)))
5		an43 671	. . 3 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
6	1, 4, 5	3bitr2ri 303	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜑)))
7		3an4anass 1122	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜑) ↔ ((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜑)))
8		ancom 466	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜑) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)))
9	6, 7, 8	3bitr2ri 303	1 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  3anasss  1380  ressply1mon1p  34033