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Theorem nfbid 1774
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 1773 . . . 4  |-  ( ph  ->  F/ x ( ps 
->  ch ) )
53, 2nfimd 1773 . . . . 5  |-  ( ph  ->  F/ x ( ch 
->  ps ) )
65nfnd 1772 . . . 4  |-  ( ph  ->  F/ x  -.  ( ch  ->  ps ) )
74, 6nfimd 1773 . . 3  |-  ( ph  ->  F/ x ( ( ps  ->  ch )  ->  -.  ( ch  ->  ps ) ) )
87nfnd 1772 . 2  |-  ( ph  ->  F/ x  -.  (
( ps  ->  ch )  ->  -.  ( ch  ->  ps ) ) )
91, 8nfxfrd 1561 1  |-  ( ph  ->  F/ x ( ps  <->  ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   F/wnf 1534
This theorem is referenced by:  nfeud2  2168  nfeqd  2446  nfiotad  5238  iota2df  5259  axextnd  8229  axrepndlem1  8230  axrepndlem2  8231  axacndlem4  8248  axacndlem5  8249  axacnd  8250  axextdist  24227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-nf 1535
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