Metamath Proof Explorer

Theorem ralrimdvv

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005)

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

Proof

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