Theorem hbimtg

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

	Ref	Expression
Assertion	hbimtg	$$⊢\left(\forall x\left(\phi \to \forall x\chi \right)\wedge \left(\psi \to \forall x\theta \right)\right)\to \left(\left(\chi \to \psi \right)\to \forall x\left(\phi \to \theta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		hbntg	$$⊢\forall x\left(\phi \to \forall x\chi \right)\to \left(\neg \chi \to \forall x\neg \phi \right)$$
2		pm2.21	$$⊢\neg \phi \to \left(\phi \to \theta \right)$$
3	2	alimi	$$⊢\forall x\neg \phi \to \forall x\left(\phi \to \theta \right)$$
4	1 3	syl6	$$⊢\forall x\left(\phi \to \forall x\chi \right)\to \left(\neg \chi \to \forall x\left(\phi \to \theta \right)\right)$$
5	4	adantr	$$⊢\left(\forall x\left(\phi \to \forall x\chi \right)\wedge \left(\psi \to \forall x\theta \right)\right)\to \left(\neg \chi \to \forall x\left(\phi \to \theta \right)\right)$$
6		ala1	$$⊢\forall x\theta \to \forall x\left(\phi \to \theta \right)$$
7	6	imim2i	$$⊢\left(\psi \to \forall x\theta \right)\to \left(\psi \to \forall x\left(\phi \to \theta \right)\right)$$
8	7	adantl	$$⊢\left(\forall x\left(\phi \to \forall x\chi \right)\wedge \left(\psi \to \forall x\theta \right)\right)\to \left(\psi \to \forall x\left(\phi \to \theta \right)\right)$$
9	5 8	jad	$$⊢\left(\forall x\left(\phi \to \forall x\chi \right)\wedge \left(\psi \to \forall x\theta \right)\right)\to \left(\left(\chi \to \psi \right)\to \forall x\left(\phi \to \theta \right)\right)$$