Theorem nfnt 1889

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 1817 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 1888	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)
2	1	biimpi 219	1 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  nfn  1890  nfnd  1891  wl-nfeqfb  38250  wl-sb8eft  38265