Theorem rspceaimv

Description: Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022)

	Ref	Expression
Hypothesis	rspceaimv.1	|- ( x = A -> ( ph <-> ps ) )
Assertion	rspceaimv	|- ( ( A e. B /\ A. y e. C ( ps -> ch ) ) -> E. x e. B A. y e. C ( ph -> ch ) )

Proof

Step	Hyp	Ref	Expression
1		rspceaimv.1	|- ( x = A -> ( ph <-> ps ) )
2	1	imbi1d	|- ( x = A -> ( ( ph -> ch ) <-> ( ps -> ch ) ) )
3	2	ralbidv	|- ( x = A -> ( A. y e. C ( ph -> ch ) <-> A. y e. C ( ps -> ch ) ) )
4	3	rspcev	|- ( ( A e. B /\ A. y e. C ( ps -> ch ) ) -> E. x e. B A. y e. C ( ph -> ch ) )