Theorem bj-zfauscl

Description: General version of zfauscl .

Remark: the comment in zfauscl is misleading: the essential use of ax-ext is the one via eleq2 and not the one via vtocl , since the latter can be proved without ax-ext (see bj-vtoclg ).

(Contributed by BJ, 2-Jul-2022) (Proof modification is discouraged.)

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

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	biimpd	$$⊢z=A\to \left(\left(x\in y\leftrightarrow \left(x\in z\wedge \phi \right)\right)\to \left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)\right)$$
5	4	alimdv	$$⊢z=A\to \left(\forall x\left(x\in y\leftrightarrow \left(x\in z\wedge \phi \right)\right)\to \forall x\left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)\right)$$
6	5	eximdv	$$⊢z=A\to \left(\exists y\forall x\left(x\in y\leftrightarrow \left(x\in z\wedge \phi \right)\right)\to \exists y\forall x\left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)\right)$$
7		ax-sep	$$⊢\exists y\forall x\left(x\in y\leftrightarrow \left(x\in z\wedge \phi \right)\right)$$
8	6 7	bj-vtoclg	$$⊢A\in V\to \exists y\forall x\left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)$$