Theorem cbvex4vw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex4v with more disjoint variable conditions, which requires fewer axioms. (Contributed by NM, 26-Jul-1995) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbvex4vw.1	|- ( ( x = v /\ y = u ) -> ( ph <-> ps ) )
	cbvex4vw.2	|- ( ( z = f /\ w = g ) -> ( ps <-> ch ) )
Assertion	cbvex4vw	|- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch )

Proof

Step	Hyp	Ref	Expression
1		cbvex4vw.1	|- ( ( x = v /\ y = u ) -> ( ph <-> ps ) )
2		cbvex4vw.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 )