Theorem ceqsexv

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995) Avoid ax-12 . (Revised by Gino Giotto, 12-Oct-2024)

	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	2	biimpa	$$⊢\left(x=A\wedge \phi \right)\to \psi$$
4	3	exlimiv	$$⊢\exists x\left(x=A\wedge \phi \right)\to \psi$$
5	2	biimprcd	$$⊢\psi \to \left(x=A\to \phi \right)$$
6	5	alrimiv	$$⊢\psi \to \forall x\left(x=A\to \phi \right)$$
7	1	isseti	$$⊢\exists xx=A$$
8		exintr	$$⊢\forall x\left(x=A\to \phi \right)\to \left(\exists xx=A\to \exists x\left(x=A\wedge \phi \right)\right)$$
9	6 7 8	mpisyl	$$⊢\psi \to \exists x\left(x=A\wedge \phi \right)$$
10	4 9	impbii	$$⊢\exists x\left(x=A\wedge \phi \right)\leftrightarrow \psi$$