Theorem biadanii 608

Description: Inference associated with biadani 607. Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.) (Proof shortened by BJ, 4-Mar-2023.)

Hypotheses

Ref	Expression
biadani.1	|- ( ph -> ps )
biadanii.2	|- ( ps -> ( ph <-> ch ) )

Assertion

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

Proof of Theorem biadanii

Step	Hyp	Ref	Expression
1		biadanii.2	. 2 |- ( ps -> ( ph <-> ch ) )
2		biadani.1	. . 3 |- ( ph -> ps )
3	2	biadani 607	. 2 |- ( ( ps -> ( ph <-> ch ) ) <-> ( ph <-> ( ps /\ ch ) ) )
4	1, 3	mpbi 144	1 |- ( ph <-> ( ps /\ ch ) )

Syntax hints:   -> wi 4   /\ 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:  ismhm  12685  iscn2  12994