Theorem anbi2ci 456

Description: Variant of anbi2i 454 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 455	. 2 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))
3		ancom 264	. 2 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓))
4	2, 3	bitri 183	1 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓))

Syntax hints:   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  clabel  2297  ordpwsucss  4551  asymref  4996  supmoti  6970  xmeter  13230