Theorem bj-cbvex4vv

Description: Version of cbvex4v with a disjoint variable condition, which does not require ax-13 . (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-cbvex4vv.1	|- ( ( x = v /\ y = u ) -> ( ph <-> ps ) )
	bj-cbvex4vv.2	|- ( ( z = f /\ w = g ) -> ( ps <-> ch ) )
Assertion	bj-cbvex4vv	|- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch )

Proof

Step	Hyp	Ref	Expression
1		bj-cbvex4vv.1	|- ( ( x = v /\ y = u ) -> ( ph <-> ps ) )
2		bj-cbvex4vv.2	|- ( ( z = f /\ w = g ) -> ( ps <-> ch ) )
3	1	2exbidv	|- ( ( x = v /\ y = u ) -> ( E. z E. w ph <-> E. z E. w ps ) )
4	3	cbvex2vw	|- ( E. x E. y E. z E. w ph <-> E. v E. u E. z E. w ps )
5	2	cbvex2vw	|- ( E. z E. w ps <-> E. f E. g ch )
6	5	2exbii	|- ( E. v E. u E. z E. w ps <-> E. v E. u E. f E. g ch )
7	4 6	bitri	|- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch )