Theorem biadani 607

Description: An implication implies to the equivalence of some implied equivalence and some other equivalence involving a conjunction. (Contributed by BJ, 4-Mar-2023.)

Hypothesis

Ref	Expression
biadani.1	|- ( ph -> ps )

Assertion

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

Proof of Theorem biadani

Step	Hyp	Ref	Expression
1		pm5.32 450	. 2 |- ( ( ps -> ( ph <-> ch ) ) <-> ( ( ps /\ ph ) <-> ( ps /\ ch ) ) )
2		biadani.1	. . . 4 |- ( ph -> ps )
3	2	pm4.71ri 390	. . 3 |- ( ph <-> ( ps /\ ph ) )
4	3	bibi1i 227	. 2 |- ( ( ph <-> ( ps /\ ch ) ) <-> ( ( ps /\ ph ) <-> ( ps /\ ch ) ) )
5	1, 4	bitr4i 186	1 |- ( ( ps -> ( ph <-> ch ) ) <-> ( 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:  biadanii  608