Theorem cbvreudavw2

Description: Change bound variable and quantifier domain in the restricted existential uniqueness quantifier. Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvreudavw2.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
	cbvreudavw2.2	|- ( ( ph /\ x = y ) -> A = B )
Assertion	cbvreudavw2	|- ( ph -> ( E! x e. A ps <-> E! y e. B ch ) )

Proof

Step	Hyp	Ref	Expression
1		cbvreudavw2.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2		cbvreudavw2.2	|- ( ( ph /\ x = y ) -> A = B )
3		simpr	|- ( ( ph /\ x = y ) -> x = y )
4	3 2	eleq12d	|- ( ( ph /\ x = y ) -> ( x e. A <-> y e. B ) )
5	4 1	anbi12d	|- ( ( ph /\ x = y ) -> ( ( x e. A /\ ps ) <-> ( y e. B /\ ch ) ) )
6	5	cbveudavw	|- ( ph -> ( E! x ( x e. A /\ ps ) <-> E! y ( y e. B /\ ch ) ) )
7		df-reu	|- ( E! x e. A ps <-> E! x ( x e. A /\ ps ) )
8		df-reu	|- ( E! y e. B ch <-> E! y ( y e. B /\ ch ) )
9	6 7 8	3bitr4g	|- ( ph -> ( E! x e. A ps <-> E! y e. B ch ) )