Theorem ceqsexgv

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996) Drop ax-10 and ax-12 . (Revised by GG, 1-Dec-2023)

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

Proof

Step	Hyp	Ref	Expression
1		ceqsexgv.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2		id	$$⊢x=A\to x=A$$
3	2 1	cgsexg	$$⊢A\in V\to \left(\exists x\left(x=A\wedge \phi \right)\leftrightarrow \psi \right)$$