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	|- ( F/ x ph <-> F/ x -. ph )

Proof of Theorem wl-nfnbi

Step	Hyp	Ref	Expression
1		nfnt 1782	. 2 |- ( F/ x ph -> F/ x -. ph )
2		notnotb 304	. . 3 |- ( ph <-> -. -. ph )
3		nfnt 1782	. . 3 |- ( F/ x -. ph -> F/ x -. -. ph )
4	2, 3	nfxfrd 1780	. 2 |- ( F/ x -. ph -> F/ x ph )
5	1, 4	impbii 199	1 |- ( F/ x ph <-> F/ x -. ph )

Syntax hints:   -. wn 3   <-> wb 196   F/wnf 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