Theorem reueq1

Description: Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023) Avoid ax-8 . (Revised by Wolf Lammen, 12-Mar-2025)

	Ref	Expression
Assertion	reueq1	$$⊢A=B\to \left(\exists !x\in A\phi \leftrightarrow \exists !x\in B\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		rexeq	$$⊢A=B\to \left(\exists x\in A\phi \leftrightarrow \exists x\in B\phi \right)$$
2		rmoeq1	$$⊢A=B\to \left({\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}x\in B\phi \right)$$
3	1 2	anbi12d	$$⊢A=B\to \left(\left(\exists x\in A\phi \wedge {\exists }^{*}x\in A\phi \right)\leftrightarrow \left(\exists x\in B\phi \wedge {\exists }^{*}x\in B\phi \right)\right)$$
4		reu5	$$⊢\exists !x\in A\phi \leftrightarrow \left(\exists x\in A\phi \wedge {\exists }^{*}x\in A\phi \right)$$
5		reu5	$$⊢\exists !x\in B\phi \leftrightarrow \left(\exists x\in B\phi \wedge {\exists }^{*}x\in B\phi \right)$$
6	3 4 5	3bitr4g	$$⊢A=B\to \left(\exists !x\in A\phi \leftrightarrow \exists !x\in B\phi \right)$$