Theorem wl-nfnbi 33314

Description: Being free does not depend on an outside negation in an expression. This theorem is slightly more general than nfn 1784 or nfnd 1785. (Contributed by Wolf Lammen, 5-May-2018.)

Assertion

Ref	Expression
wl-nfnbi	⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)

Proof of Theorem wl-nfnbi

Step	Hyp	Ref	Expression
1		nfnt 1782	. 2 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)
2		notnotb 304	. . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
3		nfnt 1782	. . 3 ⊢ (Ⅎ𝑥 ¬ 𝜑 → Ⅎ𝑥 ¬ ¬ 𝜑)
4	2, 3	nfxfrd 1780	. 2 ⊢ (Ⅎ𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑)
5	1, 4	impbii 199	1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 196  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:  wl-sb8et  33334