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	$$⊢\phi \to A=B$$
	reueqbidv.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	reueqbidv	$$⊢\phi \to \left(\exists !x\in A\psi \leftrightarrow \exists !x\in B\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		reueqbidv.1	$$⊢\phi \to A=B$$
2		reueqbidv.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
3	1	eleq2d	$$⊢\phi \to \left(x\in A\leftrightarrow x\in B\right)$$
4	3 2	anbi12d	$$⊢\phi \to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in B\wedge \chi \right)\right)$$
5	4	eubidv	$$⊢\phi \to \left(\exists !x\left(x\in A\wedge \psi \right)\leftrightarrow \exists !x\left(x\in B\wedge \chi \right)\right)$$
6		df-reu	$$⊢\exists !x\in A\psi \leftrightarrow \exists !x\left(x\in A\wedge \psi \right)$$
7		df-reu	$$⊢\exists !x\in B\chi \leftrightarrow \exists !x\left(x\in B\wedge \chi \right)$$
8	5 6 7	3bitr4g	$$⊢\phi \to \left(\exists !x\in A\psi \leftrightarrow \exists !x\in B\chi \right)$$