Theorem mpbiran3d

Description: Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 24-Sep-2024)

	Ref	Expression
Hypotheses	mpbiran3d.1	|- ( ph -> ( ps <-> ( ch /\ th ) ) )
	mpbiran3d.2	|- ( ( ph /\ ch ) -> th )
Assertion	mpbiran3d	|- ( ph -> ( ps <-> ch ) )

Proof

Step	Hyp	Ref	Expression
1		mpbiran3d.1	|- ( ph -> ( ps <-> ( ch /\ th ) ) )
2		mpbiran3d.2	|- ( ( ph /\ ch ) -> th )
3	1	simprbda	|- ( ( ph /\ ps ) -> ch )
4	3	ex	|- ( ph -> ( ps -> ch ) )
5	2	ex	|- ( ph -> ( ch -> th ) )
6	5	ancld	|- ( ph -> ( ch -> ( ch /\ th ) ) )
7	6 1	sylibrd	|- ( ph -> ( ch -> ps ) )
8	4 7	impbid	|- ( ph -> ( ps <-> ch ) )