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Metamath Proof Explorer

Theorem ralimdva

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of Margaris p. 90. (Contributed by NM, 22-May-1999) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		ralimdva.1	$⊢ (φ \land x \in A) \to (ψ \to χ)$
2	1	ex	$⊢ φ \to (x \in A \to (ψ \to χ))$
3	2	a2d	$⊢ φ \to ((x \in A \to ψ) \to (x \in A \to χ))$
4	3	ralimdv2	$⊢ φ \to (\forall x \in A ψ \to \forall x \in A χ)$