Theorem biadanii 613

Description: Inference associated with biadani 612. 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 612	. 2 |- ( ( ps -> ( ph <-> ch ) ) <-> ( ph <-> ( ps /\ ch ) ) )
4	1, 3	mpbi 145	1 |- ( ph <-> ( ps /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bitsval  12108  ismhm  13093  isghm  13373  ghmpropd  13413  isrhm  13714  iscn2  14436  elply  14970