Theorem ceqsexg

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004)

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

Proof

Step	Hyp	Ref	Expression
1		ceqsexg.1	$$⊢Ⅎx\psi$$
2		ceqsexg.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
3		nfe1	$$⊢Ⅎx\exists x\left(x=A\wedge \phi \right)$$
4	3 1	nfbi	$$⊢Ⅎx\left(\exists x\left(x=A\wedge \phi \right)\leftrightarrow \psi \right)$$
5		ceqex	$$⊢x=A\to \left(\phi \leftrightarrow \exists x\left(x=A\wedge \phi \right)\right)$$
6	5 2	bibi12d	$$⊢x=A\to \left(\left(\phi \leftrightarrow \phi \right)\leftrightarrow \left(\exists x\left(x=A\wedge \phi \right)\leftrightarrow \psi \right)\right)$$
7		biid	$$⊢\phi \leftrightarrow \phi$$
8	4 6 7	vtoclg1f	$$⊢A\in V\to \left(\exists x\left(x=A\wedge \phi \right)\leftrightarrow \psi \right)$$