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	$$⊢\phi \to \left(\psi \to \chi \right)$$
Assertion	ral2imi	$$⊢\forall x\in A\phi \to \left(\forall x\in A\psi \to \forall x\in A\chi \right)$$

Proof

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