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Theorem nfbidf 1888
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 1886 . . . 4  |-  ( ph  ->  ( A. x ps  <->  A. x ch ) )
42, 3imbi12d 320 . . 3  |-  ( ph  ->  ( ( ps  ->  A. x ps )  <->  ( ch  ->  A. x ch )
) )
51, 4albid 1886 . 2  |-  ( ph  ->  ( A. x ( ps  ->  A. x ps )  <->  A. x ( ch 
->  A. x ch )
) )
6 df-nf 1618 . 2  |-  ( F/ x ps  <->  A. x
( ps  ->  A. x ps ) )
7 df-nf 1618 . 2  |-  ( F/ x ch  <->  A. x
( ch  ->  A. x ch ) )
85, 6, 73bitr4g 288 1  |-  ( ph  ->  ( F/ x ps  <->  F/ x ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1393   F/wnf 1617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-12 1855
This theorem depends on definitions:  df-bi 185  df-ex 1614  df-nf 1618
This theorem is referenced by:  drnf2  2073  dvelimdf  2078  nfcjust  2606  nfceqdf  2614  wl-nfimf1  30162  nfbii2  30751  bj-drnf2v  34451  bj-nfcjust  34548
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