Metamath Proof Explorer

Theorem ralrimdvva

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008)

		Ref	Expression
	Hypothesis	ralrimdvva.1	$⊢ (φ \land (x \in A \land y \in B)) \to (ψ \to χ)$
	Assertion	ralrimdvva	$⊢ φ \to (ψ \to \forall x \in A \forall y \in B χ)$

Proof

Step	Hyp	Ref	Expression
1		ralrimdvva.1	$⊢ (φ \land (x \in A \land y \in B)) \to (ψ \to χ)$
2	1	ex	$⊢ φ \to ((x \in A \land y \in B) \to (ψ \to χ))$
3	2	com23	$⊢ φ \to (ψ \to ((x \in A \land y \in B) \to χ))$
4	3	ralrimdvv	$⊢ φ \to (ψ \to \forall x \in A \forall y \in B χ)$