Theorem reueqbii

Description: Equality inference for restricted existential uniqueness quantifier. (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypotheses	reueqbii.1	$$⊢A=B$$
	reueqbii.2	$$⊢\psi \leftrightarrow \chi$$
Assertion	reueqbii	$$⊢\exists !x\in A\psi \leftrightarrow \exists !x\in B\chi$$

Proof

Step	Hyp	Ref	Expression
1		reueqbii.1	$$⊢A=B$$
2		reueqbii.2	$$⊢\psi \leftrightarrow \chi$$
3	1	eleq2i	$$⊢x\in A\leftrightarrow x\in B$$
4	3 2	anbi12i	$$⊢\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in B\wedge \chi \right)$$
5	4	eubii	$$⊢\exists !x\left(x\in A\wedge \psi \right)\leftrightarrow \exists !x\left(x\in B\wedge \chi \right)$$
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	3bitr4i	$$⊢\exists !x\in A\psi \leftrightarrow \exists !x\in B\chi$$