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)

	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		eleq2	$$⊢A=B\to \left(x\in A\leftrightarrow x\in B\right)$$
2	1	anbi1d	$$⊢A=B\to \left(\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in B\wedge \phi \right)\right)$$
3	2	eubidv	$$⊢A=B\to \left(\exists !x\left(x\in A\wedge \phi \right)\leftrightarrow \exists !x\left(x\in B\wedge \phi \right)\right)$$
4		df-reu	$$⊢\exists !x\in A\phi \leftrightarrow \exists !x\left(x\in A\wedge \phi \right)$$
5		df-reu	$$⊢\exists !x\in B\phi \leftrightarrow \exists !x\left(x\in B\wedge \phi \right)$$
6	3 4 5	3bitr4g	$$⊢A=B\to \left(\exists !x\in A\phi \leftrightarrow \exists !x\in B\phi \right)$$