ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfbidf Unicode version

Theorem nfbidf 1448
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 1428 . . 3  |-  ( ph  ->  A. x ph )
3 nfbidf.2 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
42, 3albidh 1385 . . . 4  |-  ( ph  ->  ( A. x ps  <->  A. x ch ) )
53, 4imbi12d 227 . . 3  |-  ( ph  ->  ( ( ps  ->  A. x ps )  <->  ( ch  ->  A. x ch )
) )
62, 5albidh 1385 . 2  |-  ( ph  ->  ( A. x ( ps  ->  A. x ps )  <->  A. x ( ch 
->  A. x ch )
) )
7 df-nf 1366 . 2  |-  ( F/ x ps  <->  A. x
( ps  ->  A. x ps ) )
8 df-nf 1366 . 2  |-  ( F/ x ch  <->  A. x
( ch  ->  A. x ch ) )
96, 7, 83bitr4g 216 1  |-  ( ph  ->  ( F/ x ps  <->  F/ x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   A.wal 1257   F/wnf 1365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  dvelimdf  1908  nfcjust  2182  nfceqdf  2193
  Copyright terms: Public domain W3C validator