Theorem nfnt

Description: If a variable is nonfree in a proposition, then it is nonfree 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	⊢ ( Ⅎ 𝑥 𝜑 → Ⅎ 𝑥 ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		nfnbi	⊢ ( Ⅎ 𝑥 𝜑 ↔ Ⅎ 𝑥 ¬ 𝜑 )
2	1	biimpi	⊢ ( Ⅎ 𝑥 𝜑 → Ⅎ 𝑥 ¬ 𝜑 )