Theorem hbimg

Description: A more general form of hbim . (Contributed by Scott Fenton, 13-Dec-2010)

	Ref	Expression
Hypotheses	hbg.1	$$⊢\phi \to \forall x\psi$$
	hbg.2	$$⊢\chi \to \forall x\theta$$
Assertion	hbimg	$$⊢\left(\psi \to \chi \right)\to \forall x\left(\phi \to \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		hbg.1	$$⊢\phi \to \forall x\psi$$
2		hbg.2	$$⊢\chi \to \forall x\theta$$
3	1	ax-gen	$$⊢\forall x\left(\phi \to \forall x\psi \right)$$
4		hbimtg	$$⊢\left(\forall x\left(\phi \to \forall x\psi \right)\wedge \left(\chi \to \forall x\theta \right)\right)\to \left(\left(\psi \to \chi \right)\to \forall x\left(\phi \to \theta \right)\right)$$
5	3 2 4	mp2an	$$⊢\left(\psi \to \chi \right)\to \forall x\left(\phi \to \theta \right)$$