Theorem nfnt 1667

Description: If x is not free in ph, then it is not free in -. ph. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.)

Assertion

Ref	Expression
nfnt	|- ( F/ x ph -> F/ x -. ph )

Proof of Theorem nfnt

Step	Hyp	Ref	Expression
1		nfnf1 1555	. 2 |- F/ x F/ x ph
2		df-nf 1472	. . 3 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
3		hbnt 1664	. . 3 |- ( A. x ( ph -> A. x ph ) -> ( -. ph -> A. x -. ph ) )
4	2, 3	sylbi 121	. 2 |- ( F/ x ph -> ( -. ph -> A. x -. ph ) )
5	1, 4	nfd 1534	1 |- ( F/ x ph -> F/ x -. ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1362   F/wnf 1471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472

This theorem is referenced by:  nfnd  1668  nfn  1669