This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ralimdaa

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of Margaris p. 90. (Contributed by NM, 22-Sep-2003) (Proof shortened by Wolf Lammen, 29-Dec-2019)

	Ref	Expression
Hypotheses	ralimdaa.1	$⊢ Ⅎ x φ$
	ralimdaa.2	$⊢ (φ \land x \in A) \to (ψ \to χ)$
Assertion	ralimdaa	$⊢ φ \to (\forall x \in A ψ \to \forall x \in A χ)$

Proof

Step	Hyp	Ref	Expression
1		ralimdaa.1	$⊢ Ⅎ x φ$
2		ralimdaa.2	$⊢ (φ \land x \in A) \to (ψ \to χ)$
3	2	ex	$⊢ φ \to (x \in A \to (ψ \to χ))$
4	1 3	ralrimi	$⊢ φ \to \forall x \in A (ψ \to χ)$
5		ralim	$⊢ \forall x \in A (ψ \to χ) \to (\forall x \in A ψ \to \forall x \in A χ)$
6	4 5	syl	$⊢ φ \to (\forall x \in A ψ \to \forall x \in A χ)$