Theorem cbvrexvw

Description: Change the bound variable of a restricted existential quantifier using implicit substitution. Version of cbvrexv with a disjoint variable condition, which does not require ax-10 , ax-11 , ax-12 , ax-13 . (Contributed by NM, 2-Jun-1998) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypothesis	cbvralvw.1	|- ( x = y -> ( ph <-> ps ) )
Assertion	cbvrexvw	|- ( E. x e. A ph <-> E. y e. A ps )

Proof

Step	Hyp	Ref	Expression
1		cbvralvw.1	|- ( x = y -> ( ph <-> ps ) )
2		eleq1w	|- ( x = y -> ( x e. A <-> y e. A ) )
3	2 1	anbi12d	|- ( x = y -> ( ( x e. A /\ ph ) <-> ( y e. A /\ ps ) ) )
4	3	cbvexvw	|- ( E. x ( x e. A /\ ph ) <-> E. y ( y e. A /\ ps ) )
5		df-rex	|- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
6		df-rex	|- ( E. y e. A ps <-> E. y ( y e. A /\ ps ) )
7	4 5 6	3bitr4i	|- ( E. x e. A ph <-> E. y e. A ps )