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

Metamath Proof Explorer

Theorem ral2imi

Description: Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006) Allow shortening of ralim . (Revised by Wolf Lammen, 1-Dec-2019)

		Ref	Expression
	Hypothesis	ral2imi.1	$⊢ φ \to (ψ \to χ)$
	Assertion	ral2imi	$⊢ \forall x \in A φ \to (\forall x \in A ψ \to \forall x \in A χ)$

Proof

Step	Hyp	Ref	Expression
1		ral2imi.1	$⊢ φ \to (ψ \to χ)$
2		df-ral	$⊢ \forall x \in A φ \leftrightarrow \forall x (x \in A \to φ)$
3	1	imim3i	$⊢ (x \in A \to φ) \to ((x \in A \to ψ) \to (x \in A \to χ))$
4	3	al2imi	$⊢ \forall x (x \in A \to φ) \to (\forall x (x \in A \to ψ) \to \forall x (x \in A \to χ))$
5		df-ral	$⊢ \forall x \in A ψ \leftrightarrow \forall x (x \in A \to ψ)$
6		df-ral	$⊢ \forall x \in A χ \leftrightarrow \forall x (x \in A \to χ)$
7	4 5 6	3imtr4g	$⊢ \forall x (x \in A \to φ) \to (\forall x \in A ψ \to \forall x \in A χ)$
8	2 7	sylbi	$⊢ \forall x \in A φ \to (\forall x \in A ψ \to \forall x \in A χ)$