Theorem anbi2ci 446

Description: Variant of anbi2i 444 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Hypothesis

Ref	Expression
bi.aa	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
anbi2ci	⊢ ((𝜑 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓))

Proof of Theorem anbi2ci

Step	Hyp	Ref	Expression
1		bi.aa	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	anbi1i 445	. 2 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))
3		ancom 262	. 2 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓))
4	2, 3	bitri 182	1 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓))

Syntax hints:   ∧ wa 102   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  clabel  2204  ordpwsucss  4310  asymref  4730  supmoti  6406