Theorem anan 39147

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 669	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜑 ∧ 𝜃) ∧ 𝜏)) ↔ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)) ∧ (𝜒 ∧ 𝜏)))
2		anandi 689	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)))
3		ancom 466	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜃)) ↔ ((𝜓 ∧ 𝜃) ∧ 𝜑))
4	2, 3	bitr3i 280	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)) ↔ ((𝜓 ∧ 𝜃) ∧ 𝜑))
5	4	anbi1i 636	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)) ∧ (𝜒 ∧ 𝜏)) ↔ (((𝜓 ∧ 𝜃) ∧ 𝜑) ∧ (𝜒 ∧ 𝜏)))
6		anass 474	. 2 ⊢ ((((𝜓 ∧ 𝜃) ∧ 𝜑) ∧ (𝜒 ∧ 𝜏)) ↔ ((𝜓 ∧ 𝜃) ∧ (𝜑 ∧ (𝜒 ∧ 𝜏))))
7	1, 5, 6	3bitri 300	1 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ ((𝜑 ∧ 𝜃) ∧ 𝜏)) ↔ ((𝜓 ∧ 𝜃) ∧ (𝜑 ∧ (𝜒 ∧ 𝜏))))

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

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

This theorem is used by:  inxpxrn  39330