Theorem rspcedeqvd

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd for equations. (Contributed by AV, 24-Dec-2019)

	Ref	Expression
Hypotheses	rspcedeqvd.1	$$⊢\phi \to A\in B$$
	rspcedeqvd.2	$$⊢\left(\phi \wedge x=A\right)\to C=D$$
Assertion	rspcedeqvd	$$⊢\phi \to \exists x\in BC=D$$

Proof

Step	Hyp	Ref	Expression
1		rspcedeqvd.1	$$⊢\phi \to A\in B$$
2		rspcedeqvd.2	$$⊢\left(\phi \wedge x=A\right)\to C=D$$
3	2 1	rspcime	$$⊢\phi \to \exists x\in BC=D$$