Theorem reueqdv

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

	Ref	Expression
Hypothesis	reueqdv.1	$$⊢\phi \to A=B$$
Assertion	reueqdv	$$⊢\phi \to \left(\exists !x\in A\psi \leftrightarrow \exists !x\in B\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		reueqdv.1	$$⊢\phi \to A=B$$
2		reueq1	$$⊢A=B\to \left(\exists !x\in A\psi \leftrightarrow \exists !x\in B\psi \right)$$
3	1 2	syl	$$⊢\phi \to \left(\exists !x\in A\psi \leftrightarrow \exists !x\in B\psi \right)$$