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Theorem nfnt 1562
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 1452 . 2  |-  F/ x F/ x ph
2 df-nf 1366 . . 3  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
3 hbnt 1559 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  ph  ->  A. x  -.  ph )
)
42, 3sylbi 118 . 2  |-  ( F/ x ph  ->  ( -.  ph  ->  A. x  -.  ph ) )
51, 4nfd 1432 1  |-  ( F/ x ph  ->  F/ x  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1257   F/wnf 1365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366
This theorem is referenced by:  nfnd  1563  nfn  1564
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