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Theorem nfbidf 1766
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 . . 3  |-  F/ x ph
2 nfbidf.2 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
31, 2albid 1764 . . . 4  |-  ( ph  ->  ( A. x ps  <->  A. x ch ) )
42, 3imbi12d 311 . . 3  |-  ( ph  ->  ( ( ps  ->  A. x ps )  <->  ( ch  ->  A. x ch )
) )
51, 4albid 1764 . 2  |-  ( ph  ->  ( A. x ( ps  ->  A. x ps )  <->  A. x ( ch 
->  A. x ch )
) )
6 df-nf 1535 . 2  |-  ( F/ x ps  <->  A. x
( ps  ->  A. x ps ) )
7 df-nf 1535 . 2  |-  ( F/ x ch  <->  A. x
( ch  ->  A. x ch ) )
85, 6, 73bitr4g 279 1  |-  ( ph  ->  ( F/ x ps  <->  F/ x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530   F/wnf 1534
This theorem is referenced by:  nfsb4t  2033  dvelimdf  2035  nfcjust  2420  nfceqdf  2431  nfsb4twAUX7  29551  nfsb4tOLD7  29699  nfsb4tw2AUXOLD7  29700  dvelimdfOLD7  29705
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-11 1727
This theorem depends on definitions:  df-bi 177  df-nf 1535
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