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Theorem nfbidf 1477
Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)
Hypotheses
Ref Expression
nfbidf.1  |-  F/ x ph
nfbidf.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
nfbidf  |-  ( ph  ->  ( F/ x ps  <->  F/ x ch ) )

Proof of Theorem nfbidf
StepHypRef Expression
1 nfbidf.1 . . . 4  |-  F/ x ph
21nfri 1457 . . 3  |-  ( ph  ->  A. x ph )
3 nfbidf.2 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
42, 3albidh 1414 . . . 4  |-  ( ph  ->  ( A. x ps  <->  A. x ch ) )
53, 4imbi12d 232 . . 3  |-  ( ph  ->  ( ( ps  ->  A. x ps )  <->  ( ch  ->  A. x ch )
) )
62, 5albidh 1414 . 2  |-  ( ph  ->  ( A. x ( ps  ->  A. x ps )  <->  A. x ( ch 
->  A. x ch )
) )
7 df-nf 1395 . 2  |-  ( F/ x ps  <->  A. x
( ps  ->  A. x ps ) )
8 df-nf 1395 . 2  |-  ( F/ x ch  <->  A. x
( ch  ->  A. x ch ) )
96, 7, 83bitr4g 221 1  |-  ( ph  ->  ( F/ x ps  <->  F/ x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287   F/wnf 1394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  dvelimdf  1940  nfcjust  2216  nfceqdf  2227
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