Theorem hbimd

Description: Deduction form of bound-variable hypothesis builder hbim . (Contributed by NM, 14-May-1993) (Proof shortened by Wolf Lammen, 3-Jan-2018)

	Ref	Expression
Hypotheses	hbimd.1	$$⊢\phi \to \forall x\phi$$
	hbimd.2	$$⊢\phi \to \left(\psi \to \forall x\psi \right)$$
	hbimd.3	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
Assertion	hbimd	$$⊢\phi \to \left(\left(\psi \to \chi \right)\to \forall x\left(\psi \to \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		hbimd.1	$$⊢\phi \to \forall x\phi$$
2		hbimd.2	$$⊢\phi \to \left(\psi \to \forall x\psi \right)$$
3		hbimd.3	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
4	1 2	nf5dh	$$⊢\phi \to Ⅎx\psi$$
5	1 3	nf5dh	$$⊢\phi \to Ⅎx\chi$$
6	4 5	nfimd	$$⊢\phi \to Ⅎx\left(\psi \to \chi \right)$$
7	6	nf5rd	$$⊢\phi \to \left(\left(\psi \to \chi \right)\to \forall x\left(\psi \to \chi \right)\right)$$