Theorem rspcebdv

Description: Restricted existential specialization, using implicit substitution in both directions. (Contributed by AV, 8-Jan-2022)

	Ref	Expression
Hypotheses	rspcdv.1	|- ( ph -> A e. B )
	rspcdv.2	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
	rspcebdv.1	|- ( ( ph /\ ps ) -> x = A )
Assertion	rspcebdv	|- ( ph -> ( E. x e. B ps <-> ch ) )

Proof

Step	Hyp	Ref	Expression
1		rspcdv.1	|- ( ph -> A e. B )
2		rspcdv.2	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
3		rspcebdv.1	|- ( ( ph /\ ps ) -> x = A )
4	3 2	syldan	|- ( ( ph /\ ps ) -> ( ps <-> ch ) )
5	4	biimpd	|- ( ( ph /\ ps ) -> ( ps -> ch ) )
6	5	expcom	|- ( ps -> ( ph -> ( ps -> ch ) ) )
7	6	pm2.43b	|- ( ph -> ( ps -> ch ) )
8	7	rexlimdvw	|- ( ph -> ( E. x e. B ps -> ch ) )
9	1 2	rspcedv	|- ( ph -> ( ch -> E. x e. B ps ) )
10	8 9	impbid	|- ( ph -> ( E. x e. B ps <-> ch ) )