Theorem bj-cbvex2vv

Description: Version of cbvex2vv with a disjoint variable condition, which does not require ax-13 . (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	bj-cbval2vv.1	$$⊢\left(x=z\wedge y=w\right)\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	bj-cbvex2vv	$$⊢\exists x\exists y\phi \leftrightarrow \exists z\exists w\psi$$

Proof

Step	Hyp	Ref	Expression
1		bj-cbval2vv.1	$$⊢\left(x=z\wedge y=w\right)\to \left(\phi \leftrightarrow \psi \right)$$
2		nfv	$$⊢Ⅎz\phi$$
3		nfv	$$⊢Ⅎw\phi$$
4		nfv	$$⊢Ⅎx\psi$$
5		nfv	$$⊢Ⅎy\psi$$
6	2 3 4 5 1	cbvex2v	$$⊢\exists x\exists y\phi \leftrightarrow \exists z\exists w\psi$$