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

Theorem dfexdc 1431
Description: Defining  E. x ph given decidability. It is common in classical logic to define  E. x ph as  -.  A. x -.  ph but in intuitionistic logic without a decidability condition, that is only an implication not an equivalence, as seen at exalim 1432. (Contributed by Jim Kingdon, 15-Mar-2018.)
Assertion
Ref Expression
dfexdc  |-  (DECID  E. x ph  ->  ( E. x ph 
<->  -.  A. x  -.  ph ) )

Proof of Theorem dfexdc
StepHypRef Expression
1 alnex 1429 . . 3  |-  ( A. x  -.  ph  <->  -.  E. x ph )
21a1i 9 . 2  |-  (DECID  E. x ph  ->  ( A. x  -.  ph  <->  -.  E. x ph ) )
32con2biidc 807 1  |-  (DECID  E. x ph  ->  ( E. x ph 
<->  -.  A. x  -.  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103  DECID wdc 776   A.wal 1283   E.wex 1422
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-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-gen 1379  ax-ie2 1424
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-fal 1291
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator