Theorem an2anr 36064

Description: Double commutation in conjunction. (Contributed by Peter Mazsa, 27-Jun-2019.)

Assertion

Ref	Expression
an2anr	⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜓 ∧ 𝜑) ∧ (𝜃 ∧ 𝜒)))

Proof of Theorem an2anr

Step	Hyp	Ref	Expression
1		ancom 464	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
2		ancom 464	. 2 ⊢ ((𝜒 ∧ 𝜃) ↔ (𝜃 ∧ 𝜒))
3	1, 2	anbi12i 630	1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜓 ∧ 𝜑) ∧ (𝜃 ∧ 𝜒)))

Syntax hints:   ↔ wb 209   ∧ wa 399

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 210  df-an 400

This theorem is referenced by:  br1cossinres  36251  br1cossxrnres  36252