Theorem ralimdv2

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005)

	Ref	Expression
Hypothesis	ralimdv2.1	$$⊢\phi \to \left(\left(x\in A\to \psi \right)\to \left(x\in B\to \chi \right)\right)$$
Assertion	ralimdv2	$$⊢\phi \to \left(\forall x\in A\psi \to \forall x\in B\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ralimdv2.1	$$⊢\phi \to \left(\left(x\in A\to \psi \right)\to \left(x\in B\to \chi \right)\right)$$
2	1	alimdv	$$⊢\phi \to \left(\forall x\left(x\in A\to \psi \right)\to \forall x\left(x\in B\to \chi \right)\right)$$
3		df-ral	$$⊢\forall x\in A\psi \leftrightarrow \forall x\left(x\in A\to \psi \right)$$
4		df-ral	$$⊢\forall x\in B\chi \leftrightarrow \forall x\left(x\in B\to \chi \right)$$
5	2 3 4	3imtr4g	$$⊢\phi \to \left(\forall x\in A\psi \to \forall x\in B\chi \right)$$