Theorem rexbidv

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 20-Nov-1994) Reduce dependencies on axioms. (Revised by Wolf Lammen, 6-Dec-2019)

	Ref	Expression
Hypothesis	ralbidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	rexbidv	$$⊢\phi \to \left(\exists x\in A\psi \leftrightarrow \exists x\in A\chi \right)$$

Proof

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