Theorem anbi2ci 677

Description: Variant of anbi2i 675 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 676	. 2 ⊢ ((φ ∧ χ) ↔ (ψ ∧ χ))
3		ancom 437	. 2 ⊢ ((ψ ∧ χ) ↔ (χ ∧ ψ))
4	2, 3	bitri 240	1 ⊢ ((φ ∧ χ) ↔ (χ ∧ ψ))

Syntax hints:   ↔ wb 176   ∧ wa 358

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 177  df-an 360

This theorem is referenced by:  clabel  2475  enmap2lem1  6064  enmap1lem1  6070  lecex  6116