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Metamath Proof Explorer

Theorem ralimdv

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of Margaris p. 90 ( alim ). (Contributed by NM, 8-Oct-2003)

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

Proof

Step	Hyp	Ref	Expression
1		ralimdv.1	$⊢ φ \to (ψ \to χ)$
2	1	adantr	$⊢ (φ \land x \in A) \to (ψ \to χ)$
3	2	ralimdva	$⊢ φ \to (\forall x \in A ψ \to \forall x \in A χ)$