Theorem nfnt 1852

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 1779 changed. (Revised by Wolf Lammen, 4-Oct-2021.)

Assertion

Ref	Expression
nfnt	⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Proof of Theorem nfnt

Step	Hyp	Ref	Expression
1		nfnbi 1850	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)
2	1	biimpi 215	1 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804

This theorem depends on definitions:  df-bi 206  df-or 847  df-ex 1775  df-nf 1779

This theorem is referenced by:  nfn  1853  nfnd  1854  wl-nfeqfb  36998  wl-sb8eft  37013