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	$$⊢\phi \to A\in B$$
	rspcedeqvd.2	$$⊢\left(\phi \wedge x=A\right)\to C=D$$
Assertion	rspcedeq2vd	$$⊢\phi \to \exists x\in BC=D$$

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	eqcomd	$$⊢\left(\phi \wedge x=A\right)\to D=C$$
4	3	eqeq2d	$$⊢\left(\phi \wedge x=A\right)\to \left(C=D\leftrightarrow C=C\right)$$
5		eqidd	$$⊢\phi \to C=C$$
6	1 4 5	rspcedvd	$$⊢\phi \to \exists x\in BC=D$$