ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  euor2 Unicode version

Theorem euor2 2001
Description: Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
euor2  |-  ( -. 
E. x ph  ->  ( E! x ( ph  \/  ps )  <->  E! x ps ) )

Proof of Theorem euor2
StepHypRef Expression
1 hbe1 1425 . . 3  |-  ( E. x ph  ->  A. x E. x ph )
21hbn 1585 . 2  |-  ( -. 
E. x ph  ->  A. x  -.  E. x ph )
3 19.8a 1523 . . . 4  |-  ( ph  ->  E. x ph )
43con3i 595 . . 3  |-  ( -. 
E. x ph  ->  -. 
ph )
5 orel1 677 . . . 4  |-  ( -. 
ph  ->  ( ( ph  \/  ps )  ->  ps ) )
6 olc 665 . . . 4  |-  ( ps 
->  ( ph  \/  ps ) )
75, 6impbid1 140 . . 3  |-  ( -. 
ph  ->  ( ( ph  \/  ps )  <->  ps )
)
84, 7syl 14 . 2  |-  ( -. 
E. x ph  ->  ( ( ph  \/  ps ) 
<->  ps ) )
92, 8eubidh 1949 1  |-  ( -. 
E. x ph  ->  ( E! x ( ph  \/  ps )  <->  E! x ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    \/ wo 662   E.wex 1422   E!weu 1943
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-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-eu 1946
This theorem is referenced by:  reuun2  3264
  Copyright terms: Public domain W3C validator