Theorem anbi12ci 457

Description: Variant of anbi12i 456 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
anbi12.1	|- ( ph <-> ps )
anbi12.2	|- ( ch <-> th )

Assertion

Ref	Expression
anbi12ci	|- ( ( ph /\ ch ) <-> ( th /\ ps ) )

Proof of Theorem anbi12ci

Step	Hyp	Ref	Expression
1		anbi12.1	. . 3 |- ( ph <-> ps )
2		anbi12.2	. . 3 |- ( ch <-> th )
3	1, 2	anbi12i 456	. 2 |- ( ( ph /\ ch ) <-> ( ps /\ th ) )
4		ancom 264	. 2 |- ( ( ps /\ th ) <-> ( th /\ ps ) )
5	3, 4	bitri 183	1 |- ( ( ph /\ ch ) <-> ( th /\ ps ) )

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:  opelopabsbALT  4237  cnvpom  5146  f1cnvcnv  5404