Theorem 13an22anass 32483

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 636	. . 3 ⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜑)) ↔ ((𝜒 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)))
2		an2anr 636	. . . 4 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)) ↔ ((𝜒 ∧ 𝜑) ∧ (𝜓 ∧ 𝜃)))
3		an4 656	. . . 4 ⊢ (((𝜒 ∧ 𝜑) ∧ (𝜓 ∧ 𝜃)) ↔ ((𝜒 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)))
4	2, 3	bitri 275	. . 3 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)) ↔ ((𝜒 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)))
5		an43 658	. . 3 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
6	1, 4, 5	3bitr2ri 300	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜑)))
7		3an4anass 1105	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜑) ↔ ((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜑)))
8		ancom 460	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜑) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)))
9	6, 7, 8	3bitr2ri 300	1 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089

This theorem is referenced by:  ressply1mon1p  33593