Theorem reueqbidv

Description: Formula-building rule for restricted existential uniqueness quantifier. Deduction form. General version of reubidv . (Contributed by GG, 1-Sep-2025)

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

Proof

Step	Hyp	Ref	Expression
1		reueqbidv.1	|- ( ph -> A = B )
2		reueqbidv.2	|- ( ph -> ( ps <-> ch ) )
3	1	eleq2d	|- ( ph -> ( x e. A <-> x e. B ) )
4	3 2	anbi12d	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )
5	4	eubidv	|- ( ph -> ( E! x ( x e. A /\ ps ) <-> E! x ( x e. B /\ ch ) ) )
6		df-reu	|- ( E! x e. A ps <-> E! x ( x e. A /\ ps ) )
7		df-reu	|- ( E! x e. B ch <-> E! x ( x e. B /\ ch ) )
8	5 6 7	3bitr4g	|- ( ph -> ( E! x e. A ps <-> E! x e. B ch ) )