Theorem rexeqbi1dv

Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997) (Proof shortened by Steven Nguyen, 5-May-2023)

	Ref	Expression
Hypothesis	raleqbi1dv.1	$$⊢A=B\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	rexeqbi1dv	$$⊢A=B\to \left(\exists x\in A\phi \leftrightarrow \exists x\in B\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		raleqbi1dv.1	$$⊢A=B\to \left(\phi \leftrightarrow \psi \right)$$
2		id	$$⊢A=B\to A=B$$
3	2 1	rexeqbidv	$$⊢A=B\to \left(\exists x\in A\phi \leftrightarrow \exists x\in B\psi \right)$$