Theorem reueqi

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

	Ref	Expression
Hypothesis	reueqi.1	$$⊢A=B$$
Assertion	reueqi	$$⊢\exists !x\in A\psi \leftrightarrow \exists !x\in B\psi$$

Proof

Step	Hyp	Ref	Expression
1		reueqi.1	$$⊢A=B$$
2	1	eleq2i	$$⊢x\in A\leftrightarrow x\in B$$
3	2	anbi1i	$$⊢\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in B\wedge \psi \right)$$
4	3	eubii	$$⊢\exists !x\left(x\in A\wedge \psi \right)\leftrightarrow \exists !x\left(x\in B\wedge \psi \right)$$
5		df-reu	$$⊢\exists !x\in A\psi \leftrightarrow \exists !x\left(x\in A\wedge \psi \right)$$
6		df-reu	$$⊢\exists !x\in B\psi \leftrightarrow \exists !x\left(x\in B\wedge \psi \right)$$
7	4 5 6	3bitr4i	$$⊢\exists !x\in A\psi \leftrightarrow \exists !x\in B\psi$$