Theorem rexeqbidvv

Description: Version of rexeqbidv with additional disjoint variable conditions, not requiring ax-8 nor df-clel . (Contributed by Wolf Lammen, 25-Sep-2024)

	Ref	Expression
Hypotheses	raleqbidvv.1	|- ( ph -> A = B )
	raleqbidvv.2	|- ( ph -> ( ps <-> ch ) )
Assertion	rexeqbidvv	|- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )

Proof

Step	Hyp	Ref	Expression
1		raleqbidvv.1	|- ( ph -> A = B )
2		raleqbidvv.2	|- ( ph -> ( ps <-> ch ) )
3	2	notbid	|- ( ph -> ( -. ps <-> -. ch ) )
4	1 3	raleqbidvv	|- ( ph -> ( A. x e. A -. ps <-> A. x e. B -. ch ) )
5		ralnex	|- ( A. x e. A -. ps <-> -. E. x e. A ps )
6		ralnex	|- ( A. x e. B -. ch <-> -. E. x e. B ch )
7	4 5 6	3bitr3g	|- ( ph -> ( -. E. x e. A ps <-> -. E. x e. B ch ) )
8	7	con4bid	|- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )