Theorem an2anr 648

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 466	. 2 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
2		ancom 466	. 2 ⊢ ((𝜒 ∧ 𝜃) ↔ (𝜃 ∧ 𝜒))
3	1, 2	anbi12i 640	1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜓 ∧ 𝜑) ∧ (𝜃 ∧ 𝜒)))

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:  13an22anass  1379  nocvxmin  28001  br1cossinres  39246  br1cossxrnres  39247