Theorem cbvreudavw

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

	Ref	Expression
Hypothesis	cbvreudavw.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
Assertion	cbvreudavw	|- ( ph -> ( E! x e. A ps <-> E! y e. A ch ) )

Proof

Step	Hyp	Ref	Expression
1		cbvreudavw.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2		eleq1w	|- ( x = y -> ( x e. A <-> y e. A ) )
3	2	adantl	|- ( ( ph /\ x = y ) -> ( x e. A <-> y e. A ) )
4	3 1	anbi12d	|- ( ( ph /\ x = y ) -> ( ( x e. A /\ ps ) <-> ( y e. A /\ ch ) ) )
5	4	cbveudavw	|- ( ph -> ( E! x ( x e. A /\ ps ) <-> E! y ( y e. A /\ ch ) ) )
6		df-reu	|- ( E! x e. A ps <-> E! x ( x e. A /\ ps ) )
7		df-reu	|- ( E! y e. A ch <-> E! y ( y e. A /\ ch ) )
8	5 6 7	3bitr4g	|- ( ph -> ( E! x e. A ps <-> E! y e. A ch ) )