Theorem ceqsex

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995) (Revised by Mario Carneiro, 10-Oct-2016) (Proof shortened by Wolf Lammen, 22-Jan-2025)

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

Proof

Step	Hyp	Ref	Expression
1		ceqsex.1	$$⊢Ⅎx\psi$$
2		ceqsex.2	$$⊢A\in \mathrm{V}$$
3		ceqsex.3	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
4		alinexa	$$⊢\forall x\left(x=A\to \neg \phi \right)\leftrightarrow \neg \exists x\left(x=A\wedge \phi \right)$$
5	1	nfn	$$⊢Ⅎx\neg \psi$$
6	3	notbid	$$⊢x=A\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
7	5 2 6	ceqsal	$$⊢\forall x\left(x=A\to \neg \phi \right)\leftrightarrow \neg \psi$$
8	4 7	bitr3i	$$⊢\neg \exists x\left(x=A\wedge \phi \right)\leftrightarrow \neg \psi$$
9	8	con4bii	$$⊢\exists x\left(x=A\wedge \phi \right)\leftrightarrow \psi$$