Theorem 2rexbidva

Description: Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 15-Dec-2004)

	Ref	Expression
Hypothesis	2ralbidva.1	$$⊢\left(\phi \wedge \left(x\in A\wedge y\in B\right)\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	2rexbidva	$$⊢\phi \to \left(\exists x\in A\exists y\in B\psi \leftrightarrow \exists x\in A\exists y\in B\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		2ralbidva.1	$$⊢\left(\phi \wedge \left(x\in A\wedge y\in B\right)\right)\to \left(\psi \leftrightarrow \chi \right)$$
2	1	anassrs	$$⊢\left(\left(\phi \wedge x\in A\right)\wedge y\in B\right)\to \left(\psi \leftrightarrow \chi \right)$$
3	2	rexbidva	$$⊢\left(\phi \wedge x\in A\right)\to \left(\exists y\in B\psi \leftrightarrow \exists y\in B\chi \right)$$
4	3	rexbidva	$$⊢\phi \to \left(\exists x\in A\exists y\in B\psi \leftrightarrow \exists x\in A\exists y\in B\chi \right)$$