Theorem nfnt 1782

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 1710 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 1781	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)
2	1	biimpi 206	1 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by:  nfn  1784  nfnd  1785  nfan1  2068  19.23t  2079  wl-nfnbi  33314  wl-nfeqfb  33323  19.9alt  34252