Theorem rspceeqv

Description: Restricted existential specialization in an equality, using implicit substitution. (Contributed by BJ, 2-Sep-2022)

	Ref	Expression
Hypothesis	rspceeqv.1	$$⊢x=A\to C=D$$
Assertion	rspceeqv	$$⊢\left(A\in B\wedge E=D\right)\to \exists x\in BE=C$$

Proof

Step	Hyp	Ref	Expression
1		rspceeqv.1	$$⊢x=A\to C=D$$
2	1	eqeq2d	$$⊢x=A\to \left(E=C\leftrightarrow E=D\right)$$
3	2	rspcev	$$⊢\left(A\in B\wedge E=D\right)\to \exists x\in BE=C$$