Theorem cbvrexvw2

Description: Change bound variable and domain in the restricted existential quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvrexvw2.1	|- ( x = y -> A = B )
	cbvrexvw2.2	|- ( x = y -> ( ph <-> ps ) )
Assertion	cbvrexvw2	|- ( E. x e. A ph <-> E. y e. B ps )

Proof

Step	Hyp	Ref	Expression
1		cbvrexvw2.1	|- ( x = y -> A = B )
2		cbvrexvw2.2	|- ( x = y -> ( ph <-> ps ) )
3		eleq1w	|- ( x = y -> ( x e. A <-> y e. A ) )
4	1	eleq2d	|- ( x = y -> ( y e. A <-> y e. B ) )
5	3 4	bitrd	|- ( x = y -> ( x e. A <-> y e. B ) )
6	5 2	anbi12d	|- ( x = y -> ( ( x e. A /\ ph ) <-> ( y e. B /\ ps ) ) )
7	6	cbvexvw	|- ( E. x ( x e. A /\ ph ) <-> E. y ( y e. B /\ ps ) )
8		df-rex	|- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
9		df-rex	|- ( E. y e. B ps <-> E. y ( y e. B /\ ps ) )
10	7 8 9	3bitr4i	|- ( E. x e. A ph <-> E. y e. B ps )