Theorem rspcev

Description: Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998) Drop ax-10 , ax-11 , ax-12 . (Revised by SN, 12-Dec-2023)

	Ref	Expression
Hypothesis	rspcv.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	rspcev	$$⊢\left(A\in B\wedge \psi \right)\to \exists x\in B\phi$$

Proof

Step	Hyp	Ref	Expression
1		rspcv.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2		id	$$⊢A\in B\to A\in B$$
3	1	adantl	$$⊢\left(A\in B\wedge x=A\right)\to \left(\phi \leftrightarrow \psi \right)$$
4	2 3	rspcedv	$$⊢A\in B\to \left(\psi \to \exists x\in B\phi \right)$$
5	4	imp	$$⊢\left(A\in B\wedge \psi \right)\to \exists x\in B\phi$$