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) (Proof shortened by Wolf Lammen, 22-Jan-2025)

	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		alinexa	$$⊢\forall x\left(x=A\to \neg \phi \right)\leftrightarrow \neg \exists x\left(x=A\wedge \phi \right)$$
4	2	notbid	$$⊢x=A\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
5	1 4	ceqsalv	$$⊢\forall x\left(x=A\to \neg \phi \right)\leftrightarrow \neg \psi$$
6	3 5	bitr3i	$$⊢\neg \exists x\left(x=A\wedge \phi \right)\leftrightarrow \neg \psi$$
7	6	con4bii	$$⊢\exists x\left(x=A\wedge \phi \right)\leftrightarrow \psi$$