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Metamath Proof Explorer

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	$⊢ φ \to A \in B$
	rspcedeqvd.2	$⊢ (φ \land x = A) \to C = D$
Assertion	rspcedeq2vd	$⊢ φ \to \exists x \in B C = D$

Proof

Step	Hyp	Ref	Expression
1		rspcedeqvd.1	$⊢ φ \to A \in B$
2		rspcedeqvd.2	$⊢ (φ \land x = A) \to C = D$
3	2	eqcomd	$⊢ (φ \land x = A) \to D = C$
4	3	eqeq2d	$⊢ (φ \land x = A) \to (C = D \leftrightarrow C = C)$
5		eqidd	$⊢ φ \to C = C$
6	1 4 5	rspcedvd	$⊢ φ \to \exists x \in B C = D$