Theorem sepg

Description: Version of the axiom of separation where the "containing set" is a class variable and the sethood assumption is in the antecedent. (Contributed by NM, 21-Jun-1993) Put sepgi in closed form. (Revised by BJ, 2-Jul-2022)

	Ref	Expression
Assertion	sepg	$$⊢A\in V\to \exists y\forall x\left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		eleq2	$$⊢z=A\to \left(x\in z\leftrightarrow x\in A\right)$$
2	1	anbi1d	$$⊢z=A\to \left(\left(x\in z\wedge \phi \right)\leftrightarrow \left(x\in A\wedge \phi \right)\right)$$
3	2	bibi2d	$$⊢z=A\to \left(\left(x\in y\leftrightarrow \left(x\in z\wedge \phi \right)\right)\leftrightarrow \left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)\right)$$
4	3	albidv	$$⊢z=A\to \left(\forall x\left(x\in y\leftrightarrow \left(x\in z\wedge \phi \right)\right)\leftrightarrow \forall x\left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)\right)$$
5	4	exbidv	$$⊢z=A\to \left(\exists y\forall x\left(x\in y\leftrightarrow \left(x\in z\wedge \phi \right)\right)\leftrightarrow \exists y\forall x\left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)\right)$$
6		ax-sep	$$⊢\exists y\forall x\left(x\in y\leftrightarrow \left(x\in z\wedge \phi \right)\right)$$
7	5 6	vtoclg	$$⊢A\in V\to \exists y\forall x\left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)$$