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Theorem nfbid 1844
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 1817 . . 3  |-  ( ph  ->  F/ x ( ps 
->  ch ) )
53, 2nfimd 1817 . . 3  |-  ( ph  ->  F/ x ( ch 
->  ps ) )
64, 5nfand 1833 . 2  |-  ( ph  ->  F/ x ( ( ps  ->  ch )  /\  ( ch  ->  ps ) ) )
71, 6nfxfrd 1577 1  |-  ( ph  ->  F/ x ( ps  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1550
This theorem is referenced by:  nfbi  1846  nfeud2  2250  nfeqd  2537  nfiotad  5361  iota2df  5382  axextnd  8399  axrepndlem1  8400  axrepndlem2  8401  axacndlem4  8418  axacndlem5  8419  axacnd  8420  axextdist  25180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
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