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Metamath Proof Explorer

Theorem ralimdv2

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005)

		Ref	Expression
	Hypothesis	ralimdv2.1	$⊢ φ \to ((x \in A \to ψ) \to (x \in B \to χ))$
	Assertion	ralimdv2	$⊢ φ \to (\forall x \in A ψ \to \forall x \in B χ)$

Step	Hyp	Ref	Expression
1		ralimdv2.1	$⊢ φ \to ((x \in A \to ψ) \to (x \in B \to χ))$
2	1	alimdv	$⊢ φ \to (\forall x (x \in A \to ψ) \to \forall x (x \in B \to χ))$
3		df-ral	$⊢ \forall x \in A ψ \leftrightarrow \forall x (x \in A \to ψ)$
4		df-ral	$⊢ \forall x \in B χ \leftrightarrow \forall x (x \in B \to χ)$
5	2 3 4	3imtr4g	$⊢ φ \to (\forall x \in A ψ \to \forall x \in B χ)$