Theorem anbi2ci 732

Description: Variant of anbi2i 730 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Hypothesis

Ref	Expression
anbi.1	⊢ (𝜑 ↔ 𝜓)

Assertion

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

Proof of Theorem anbi2ci

Step	Hyp	Ref	Expression
1		anbi.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	anbi1i 731	. 2 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))
3		ancom 466	. 2 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓))
4	2, 3	bitri 264	1 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓))

Syntax hints:   ↔ wb 196   ∧ wa 384

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 197  df-an 386

This theorem is referenced by:  ifpn  1022  clabel  2749  difin0ss  3946  disjxun  4651  asymref  5512  ordpwsuc  7015  supmo  8358  infmo  8401  kmlem3  8974  cfval2  9082  eqger  17644  gaorber  17741  opprunit  18661  xmeter  22238  iscvsp  22928  usgr2pth0  26661  bj-clabel  32783  clsk1indlem4  38342  alimp-no-surprise  42527