Theorem hbnt

Description: Closed theorem version of bound-variable hypothesis builder hbn . (Contributed by NM, 10-May-1993) (Proof shortened by Wolf Lammen, 14-Oct-2021)

	Ref	Expression
Assertion	hbnt	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to \left(\neg \phi \to \forall x\neg \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nf5-1	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to Ⅎx\phi$$
2	1	nfnd	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to Ⅎx\neg \phi$$
3	2	nf5rd	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to \left(\neg \phi \to \forall x\neg \phi \right)$$