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Theorem nfnd 1809
Description: If in a context  x is not free in  ps, it is not free in  -.  ps. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.)
Hypothesis
Ref Expression
nfnd.1  |-  ( ph  ->  F/ x ps )
Assertion
Ref Expression
nfnd  |-  ( ph  ->  F/ x  -.  ps )

Proof of Theorem nfnd
StepHypRef Expression
1 nfnd.1 . 2  |-  ( ph  ->  F/ x ps )
2 nfnf1 1808 . . 3  |-  F/ x F/ x ps
3 df-nf 1554 . . . 4  |-  ( F/ x ps  <->  A. x
( ps  ->  A. x ps ) )
4 hbnt 1799 . . . 4  |-  ( A. x ( ps  ->  A. x ps )  -> 
( -.  ps  ->  A. x  -.  ps )
)
53, 4sylbi 188 . . 3  |-  ( F/ x ps  ->  ( -.  ps  ->  A. x  -.  ps ) )
62, 5nfd 1782 . 2  |-  ( F/ x ps  ->  F/ x  -.  ps )
71, 6syl 16 1  |-  ( ph  ->  F/ x  -.  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549   F/wnf 1553
This theorem is referenced by:  nfn  1811  nfand  1843  nfbidOLD  1855  nfexd  1873  19.9tOLD  1880  cbvexd  1988  nfexd2  2065  nfned  2696  nfneld  2703  nfrexd  2758  axpowndlem2  8473  axpowndlem3  8474  axpowndlem4  8475  axregndlem2  8478  axregnd  8479  distel  25431  wl-nfnbi  26233  nfexdwAUX7  29453  nfexdOLD7  29691  nfexd2OLD7  29718  cbvexdOLD7  29735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551  df-nf 1554
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