Theorem anan 33999

Description: Multiple commutations in conjunction. (Contributed by Peter Mazsa, 7-Mar-2020.)

Assertion

Ref	Expression
anan	⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜑 ∧ 𝜃) ∧ 𝜏)) ↔ ((𝜓 ∧ 𝜃) ∧ (𝜑 ∧ (𝜒 ∧ 𝜏))))

Proof of Theorem anan

Step	Hyp	Ref	Expression
1		an4 865	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜑 ∧ 𝜃) ∧ 𝜏)) ↔ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)) ∧ (𝜒 ∧ 𝜏)))
2		anandi 871	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)))
3		ancom 466	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜃)) ↔ ((𝜓 ∧ 𝜃) ∧ 𝜑))
4	2, 3	bitr3i 266	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)) ↔ ((𝜓 ∧ 𝜃) ∧ 𝜑))
5	4	anbi1i 731	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)) ∧ (𝜒 ∧ 𝜏)) ↔ (((𝜓 ∧ 𝜃) ∧ 𝜑) ∧ (𝜒 ∧ 𝜏)))
6		anass 681	. 2 ⊢ ((((𝜓 ∧ 𝜃) ∧ 𝜑) ∧ (𝜒 ∧ 𝜏)) ↔ ((𝜓 ∧ 𝜃) ∧ (𝜑 ∧ (𝜒 ∧ 𝜏))))
7	1, 5, 6	3bitri 286	1 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜑 ∧ 𝜃) ∧ 𝜏)) ↔ ((𝜓 ∧ 𝜃) ∧ (𝜑 ∧ (𝜒 ∧ 𝜏))))

Syntax hints:   ↔ wb 196   ∧ wa 384

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 197  df-an 386

This theorem is referenced by: (None)