Theorem ceqsex

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995) (Revised by Mario Carneiro, 10-Oct-2016)

	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	3	biimpa	$$⊢\left(x=A\wedge \phi \right)\to \psi$$
5	1 4	exlimi	$$⊢\exists x\left(x=A\wedge \phi \right)\to \psi$$
6	3	biimprcd	$$⊢\psi \to \left(x=A\to \phi \right)$$
7	1 6	alrimi	$$⊢\psi \to \forall x\left(x=A\to \phi \right)$$
8	2	isseti	$$⊢\exists xx=A$$
9		exintr	$$⊢\forall x\left(x=A\to \phi \right)\to \left(\exists xx=A\to \exists x\left(x=A\wedge \phi \right)\right)$$
10	7 8 9	mpisyl	$$⊢\psi \to \exists x\left(x=A\wedge \phi \right)$$
11	5 10	impbii	$$⊢\exists x\left(x=A\wedge \phi \right)\leftrightarrow \psi$$