Theorem reueqbidva

Description: Formula-building rule for restricted existential uniqueness quantifier. Deduction form. General version of reueqbidv . (Contributed by Zhi Wang, 20-Nov-2025)

	Ref	Expression
Hypotheses	reueqbidva.1	$$⊢\phi \to A=B$$
	reueqbidva.2	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	reueqbidva	$$⊢\phi \to \left(\exists !x\in A\psi \leftrightarrow \exists !x\in B\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		reueqbidva.1	$$⊢\phi \to A=B$$
2		reueqbidva.2	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3	2	reubidva	$$⊢\phi \to \left(\exists !x\in A\psi \leftrightarrow \exists !x\in A\chi \right)$$
4	1	reueqdv	$$⊢\phi \to \left(\exists !x\in A\chi \leftrightarrow \exists !x\in B\chi \right)$$
5	3 4	bitrd	$$⊢\phi \to \left(\exists !x\in A\psi \leftrightarrow \exists !x\in B\chi \right)$$