Theorem rspcedeq2vd

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd for equations, in which the right hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019)

	Ref	Expression
Hypotheses	rspcedeqvd.1	|- ( ph -> A e. B )
	rspcedeqvd.2	|- ( ( ph /\ x = A ) -> C = D )
Assertion	rspcedeq2vd	|- ( ph -> E. x e. B C = D )

Proof

Step	Hyp	Ref	Expression
1		rspcedeqvd.1	|- ( ph -> A e. B )
2		rspcedeqvd.2	|- ( ( ph /\ x = A ) -> C = D )
3	2	eqcomd	|- ( ( ph /\ x = A ) -> D = C )
4	3	eqeq2d	|- ( ( ph /\ x = A ) -> ( C = D <-> C = C ) )
5		eqidd	|- ( ph -> C = C )
6	1 4 5	rspcedvd	|- ( ph -> E. x e. B C = D )