Theorem nfnt

Description: If a variable is non-free in a proposition, then it is non-free in its negation. (Contributed by Mario Carneiro, 24-Sep-2016) (Proof shortened by Wolf Lammen, 28-Dec-2017) (Revised by BJ, 24-Jul-2019) df-nf changed. (Revised by Wolf Lammen, 4-Oct-2021)

	Ref	Expression
Assertion	nfnt	$$⊢Ⅎx\phi \to Ⅎx\neg \phi$$

Proof

Step	Hyp	Ref	Expression
1		nfnbi	$$⊢Ⅎx\phi \leftrightarrow Ⅎx\neg \phi$$
2	1	biimpi	$$⊢Ⅎx\phi \to Ⅎx\neg \phi$$