Theorem ralrimivv

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004)

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

Proof

Step	Hyp	Ref	Expression
1		ralrimivv.1	$$⊢\phi \to \left(\left(x\in A\wedge y\in B\right)\to \psi \right)$$
2	1	expd	$$⊢\phi \to \left(x\in A\to \left(y\in B\to \psi \right)\right)$$
3	2	ralrimdv	$$⊢\phi \to \left(x\in A\to \forall y\in B\psi \right)$$
4	3	ralrimiv	$$⊢\phi \to \forall x\in A\forall y\in B\psi$$