Theorem nfnt 1708

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 1597	. 2 |- F/ x F/ x ph
2		df-nf 1514	. . 3 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
3		hbnt 1705	. . 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 1576	1 |- ( F/ x ph -> F/ x -. ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1400   F/wnf 1513

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 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie2 1547  ax-4 1563  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514

This theorem is referenced by:  nfnd  1709  nfn  1710