Theorem mpbiran2d 442

Description: Detach truth from conjunction in biconditional. Deduction form. (Contributed by Peter Mazsa, 24-Sep-2022.)

Hypotheses

Ref	Expression
mpbiran2d.1	|- ( ph -> th )
mpbiran2d.2	|- ( ph -> ( ps <-> ( ch /\ th ) ) )

Assertion

Ref	Expression
mpbiran2d	|- ( ph -> ( ps <-> ch ) )

Proof of Theorem mpbiran2d

Step	Hyp	Ref	Expression
1		mpbiran2d.2	. 2 |- ( ph -> ( ps <-> ( ch /\ th ) ) )
2		mpbiran2d.1	. . 3 |- ( ph -> th )
3	2	biantrud 304	. 2 |- ( ph -> ( ch <-> ( ch /\ th ) ) )
4	1, 3	bitr4d 191	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:  lgsneg  14296