Theorem rexeqf

Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See rexeq for a version based on fewer axioms. (Contributed by NM, 9-Oct-2003) (Revised by Andrew Salmon, 11-Jul-2011) (Proof shortened by Wolf Lammen, 9-Mar-2025)

	Ref	Expression
Hypotheses	raleqf.1	$$⊢\underset{\_}{Ⅎ}xA$$
	raleqf.2	$$⊢\underset{\_}{Ⅎ}xB$$
Assertion	rexeqf	$$⊢A=B\to \left(\exists x\in A\phi \leftrightarrow \exists x\in B\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		raleqf.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		raleqf.2	$$⊢\underset{\_}{Ⅎ}xB$$
3	1 2	raleqf	$$⊢A=B\to \left(\forall x\in A\neg \phi \leftrightarrow \forall x\in B\neg \phi \right)$$
4		ralnex	$$⊢\forall x\in A\neg \phi \leftrightarrow \neg \exists x\in A\phi$$
5		ralnex	$$⊢\forall x\in B\neg \phi \leftrightarrow \neg \exists x\in B\phi$$
6	3 4 5	3bitr3g	$$⊢A=B\to \left(\neg \exists x\in A\phi \leftrightarrow \neg \exists x\in B\phi \right)$$
7	6	con4bid	$$⊢A=B\to \left(\exists x\in A\phi \leftrightarrow \exists x\in B\phi \right)$$