Theorem rexbid

Description: Formula-building rule for restricted existential quantifier (deduction form). For a version based on fewer axioms see rexbidv . (Contributed by NM, 27-Jun-1998)

	Ref	Expression
Hypotheses	rexbid.1	$$⊢Ⅎx\phi$$
	rexbid.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	rexbid	$$⊢\phi \to \left(\exists x\in A\psi \leftrightarrow \exists x\in A\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		rexbid.1	$$⊢Ⅎx\phi$$
2		rexbid.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
3	2	adantr	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
4	1 3	rexbida	$$⊢\phi \to \left(\exists x\in A\psi \leftrightarrow \exists x\in A\chi \right)$$