Theorem sepgi

Description: Inference associated with sepg . The requirement that y not occur in ph is necessary, as notsep shows. (Contributed by NM, 21-Jun-1993) (Revised by BJ, 14-Jul-2026)

	Ref	Expression
Hypothesis	sepgi.1	$$⊢A\in \mathrm{V}$$
Assertion	sepgi	$$⊢\exists y\forall x\left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sepgi.1	$$⊢A\in \mathrm{V}$$
2		sepg	$$⊢A\in \mathrm{V}\to \exists y\forall x\left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)$$
3	1 2	ax-mp	$$⊢\exists y\forall x\left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)$$