Theorem ceqsexv2d

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016)

	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 2	ceqsexv	$$⊢\exists x\left(x=A\wedge \phi \right)\leftrightarrow \psi$$
5	4	biimpri	$$⊢\psi \to \exists x\left(x=A\wedge \phi \right)$$
6		exsimpr	$$⊢\exists x\left(x=A\wedge \phi \right)\to \exists x\phi$$
7	3 5 6	mp2b	$$⊢\exists x\phi$$