MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfbidf Unicode version

Theorem nfbidf 1754
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 1752 . . . 4  |-  ( ph  ->  ( A. x ps  <->  A. x ch ) )
42, 3imbi12d 311 . . 3  |-  ( ph  ->  ( ( ps  ->  A. x ps )  <->  ( ch  ->  A. x ch )
) )
51, 4albid 1752 . 2  |-  ( ph  ->  ( A. x ( ps  ->  A. x ps )  <->  A. x ( ch 
->  A. x ch )
) )
6 df-nf 1532 . 2  |-  ( F/ x ps  <->  A. x
( ps  ->  A. x ps ) )
7 df-nf 1532 . 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 1527   F/wnf 1531
This theorem is referenced by:  nfsb4t  2020  dvelimdf  2022  nfcjust  2407  nfceqdf  2418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-nf 1532
  Copyright terms: Public domain W3C validator