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Theorem nfbid 1762
Description: If  x is not free in  ph and  ps, it is not free in  ( ph  ->  ps ). (Contributed by Mario Carneiro, 24-Sep-2016.)
Hypotheses
Ref Expression
nfnd.1  |-  ( ph  ->  F/ x ps )
nfimd.2  |-  ( ph  ->  F/ x ch )
Assertion
Ref Expression
nfbid  |-  ( ph  ->  F/ x ( ps  <->  ch ) )

Proof of Theorem nfbid
StepHypRef Expression
1 dfbi1 184 . 2  |-  ( ( ps  <->  ch )  <->  -.  (
( ps  ->  ch )  ->  -.  ( ch  ->  ps ) ) )
2 nfnd.1 . . . . 5  |-  ( ph  ->  F/ x ps )
3 nfimd.2 . . . . 5  |-  ( ph  ->  F/ x ch )
42, 3nfimd 1761 . . . 4  |-  ( ph  ->  F/ x ( ps 
->  ch ) )
53, 2nfimd 1761 . . . . 5  |-  ( ph  ->  F/ x ( ch 
->  ps ) )
65nfnd 1760 . . . 4  |-  ( ph  ->  F/ x  -.  ( ch  ->  ps ) )
74, 6nfimd 1761 . . 3  |-  ( ph  ->  F/ x ( ( ps  ->  ch )  ->  -.  ( ch  ->  ps ) ) )
87nfnd 1760 . 2  |-  ( ph  ->  F/ x  -.  (
( ps  ->  ch )  ->  -.  ( ch  ->  ps ) ) )
91, 8nfxfrd 1558 1  |-  ( ph  ->  F/ x ( ps  <->  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   F/wnf 1531
This theorem is referenced by:  nfeud2  2155  nfeqd  2433  nfiotad  5222  iota2df  5243  axextnd  8213  axrepndlem1  8214  axrepndlem2  8215  axacndlem4  8232  axacndlem5  8233  axacnd  8234  axextdist  24156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-nf 1532
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