Theorem rexlimdvva

Description: Inference from Theorem 19.23 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014)

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

Proof

Step	Hyp	Ref	Expression
1		rexlimdvva.1	$$⊢\left(\phi \wedge \left(x\in A\wedge y\in B\right)\right)\to \left(\psi \to \chi \right)$$
2	1	ex	$$⊢\phi \to \left(\left(x\in A\wedge y\in B\right)\to \left(\psi \to \chi \right)\right)$$
3	2	rexlimdvv	$$⊢\phi \to \left(\exists x\in A\exists y\in B\psi \to \chi \right)$$