Theorem ceqsexv2d

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016) Shorten, reduce dv conditions. (Revised by Wolf Lammen, 5-Jun-2025) (Proof shortened by SN, 5-Jun-2025)

	Ref	Expression
Hypotheses	ceqsexv2d.1	$$⊢A\in \mathrm{V}$$
	ceqsexv2d.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	ceqsexv2d.3	$$⊢\psi$$
Assertion	ceqsexv2d	$$⊢\exists x\phi$$

Proof

Step	Hyp	Ref	Expression
1		ceqsexv2d.1	$$⊢A\in \mathrm{V}$$
2		ceqsexv2d.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
3		ceqsexv2d.3	$$⊢\psi$$
4	1	isseti	$$⊢\exists xx=A$$
5	3 2	mpbiri	$$⊢x=A\to \phi$$
6	4 5	eximii	$$⊢\exists x\phi$$