Theorem rexbidva

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 9-Mar-1997) Reduce dependencies on axioms. (Revised by Wolf Lammen, 6-Dec-2019) (Proof shortened by Wolf Lammen, 10-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		ralbidva.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
2	1	pm5.32da	$$⊢\phi \to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in A\wedge \chi \right)\right)$$
3	2	rexbidv2	$$⊢\phi \to \left(\exists x\in A\psi \leftrightarrow \exists x\in A\chi \right)$$