Theorem hbn

Description: If x is not free in ph , it is not free in -. ph . (Contributed by NM, 10-Jan-1993) (Proof shortened by Wolf Lammen, 17-Dec-2017)

	Ref	Expression
Hypothesis	hbn.1	$$⊢\phi \to \forall x\phi$$
Assertion	hbn	$$⊢\neg \phi \to \forall x\neg \phi$$

Proof

Step	Hyp	Ref	Expression
1		hbn.1	$$⊢\phi \to \forall x\phi$$
2		hbnt	$$⊢\forall x\left(\phi \to \forall x\phi \right)\to \left(\neg \phi \to \forall x\neg \phi \right)$$
3	2 1	mpg	$$⊢\neg \phi \to \forall x\neg \phi$$