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Theorem nfbid 1854
Description: If in a context  x is not free in  ps and  ch, it is not free in  ( ps  <->  ch ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)
Hypotheses
Ref Expression
nfbid.1  |-  ( ph  ->  F/ x ps )
nfbid.2  |-  ( ph  ->  F/ x ch )
Assertion
Ref Expression
nfbid  |-  ( ph  ->  F/ x ( ps  <->  ch ) )

Proof of Theorem nfbid
StepHypRef Expression
1 dfbi2 610 . 2  |-  ( ( ps  <->  ch )  <->  ( ( ps  ->  ch )  /\  ( ch  ->  ps )
) )
2 nfbid.1 . . . 4  |-  ( ph  ->  F/ x ps )
3 nfbid.2 . . . 4  |-  ( ph  ->  F/ x ch )
42, 3nfimd 1827 . . 3  |-  ( ph  ->  F/ x ( ps 
->  ch ) )
53, 2nfimd 1827 . . 3  |-  ( ph  ->  F/ x ( ch 
->  ps ) )
64, 5nfand 1843 . 2  |-  ( ph  ->  F/ x ( ( ps  ->  ch )  /\  ( ch  ->  ps ) ) )
71, 6nfxfrd 1580 1  |-  ( ph  ->  F/ x ( ps  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1553
This theorem is referenced by:  nfbi  1856  nfeud2  2292  nfeqd  2585  nfiotad  5413  iota2df  5434  axextnd  8458  axrepndlem1  8459  axrepndlem2  8460  axacndlem4  8477  axacndlem5  8478  axacnd  8479  axextdist  25419
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-an 361  df-ex 1551  df-nf 1554
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