Theorem rexbida

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 6-Oct-2003)

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

Proof

Step	Hyp	Ref	Expression
1		rexbida.1	$$⊢Ⅎx\phi$$
2		rexbida.2	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3	2	pm5.32da	$$⊢\phi \to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in A\wedge \chi \right)\right)$$
4	1 3	exbid	$$⊢\phi \to \left(\exists x\left(x\in A\wedge \psi \right)\leftrightarrow \exists x\left(x\in A\wedge \chi \right)\right)$$
5		df-rex	$$⊢\exists x\in A\psi \leftrightarrow \exists x\left(x\in A\wedge \psi \right)$$
6		df-rex	$$⊢\exists x\in A\chi \leftrightarrow \exists x\left(x\in A\wedge \chi \right)$$
7	4 5 6	3bitr4g	$$⊢\phi \to \left(\exists x\in A\psi \leftrightarrow \exists x\in A\chi \right)$$