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Theorem nfnt 1959
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
StepHypRef Expression
1 nfnf1 1958 . 2  |-  F/ x F/ x ph
2 df-nf 1662 . . 3  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
3 hbnt 1953 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  ph  ->  A. x  -.  ph )
)
42, 3sylbi 198 . 2  |-  ( F/ x ph  ->  ( -.  ph  ->  A. x  -.  ph ) )
51, 4nfd 1933 1  |-  ( F/ x ph  ->  F/ x  -.  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1435   F/wnf 1661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909
This theorem depends on definitions:  df-bi 188  df-ex 1658  df-nf 1662
This theorem is referenced by:  nfn  1960  nfnd  1961  19.23t  1968  wl-nfnbi  31823  wl-nfeqfb  31834
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