Theorem ceqsexv

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995)

	Ref	Expression
Hypotheses	ceqsexv.1	$$⊢A\in \mathrm{V}$$
	ceqsexv.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	ceqsexv	$$⊢\exists x\left(x=A\wedge \phi \right)\leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		ceqsexv.1	$$⊢A\in \mathrm{V}$$
2		ceqsexv.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
3		nfv	$$⊢Ⅎx\psi$$
4	3 1 2	ceqsex	$$⊢\exists x\left(x=A\wedge \phi \right)\leftrightarrow \psi$$