Theorem rspcedeq1vd

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd for equations, in which the left hand side depends on the quantified variable. (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	rspcedeq1vd	$$⊢\phi \to \exists x\in BC=D$$

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	eqeq1d	$$⊢\left(\phi \wedge x=A\right)\to \left(C=D\leftrightarrow D=D\right)$$
4		eqidd	$$⊢\phi \to D=D$$
5	1 3 4	rspcedvd	$$⊢\phi \to \exists x\in BC=D$$