Theorem rexeq

Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023) Shorten other proofs. (Revised by Wolf Lammen, 8-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		dfcleq	$$⊢A=B\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in B\right)$$
2		anbi1	$$⊢\left(x\in A\leftrightarrow x\in B\right)\to \left(\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in B\wedge \phi \right)\right)$$
3	2	alexbii	$$⊢\forall x\left(x\in A\leftrightarrow x\in B\right)\to \left(\exists x\left(x\in A\wedge \phi \right)\leftrightarrow \exists x\left(x\in B\wedge \phi \right)\right)$$
4	1 3	sylbi	$$⊢A=B\to \left(\exists x\left(x\in A\wedge \phi \right)\leftrightarrow \exists x\left(x\in B\wedge \phi \right)\right)$$
5		df-rex	$$⊢\exists x\in A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$
6		df-rex	$$⊢\exists x\in B\phi \leftrightarrow \exists x\left(x\in B\wedge \phi \right)$$
7	4 5 6	3bitr4g	$$⊢A=B\to \left(\exists x\in A\phi \leftrightarrow \exists x\in B\phi \right)$$