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Theorem euor2 2006
Description: Introduce or eliminate a disjunct in a unique existential 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 1429 . . 3  |-  ( E. x ph  ->  A. x E. x ph )
21hbn 1589 . 2  |-  ( -. 
E. x ph  ->  A. x  -.  E. x ph )
3 19.8a 1527 . . . 4  |-  ( ph  ->  E. x ph )
43con3i 597 . . 3  |-  ( -. 
E. x ph  ->  -. 
ph )
5 orel1 679 . . . 4  |-  ( -. 
ph  ->  ( ( ph  \/  ps )  ->  ps ) )
6 olc 667 . . . 4  |-  ( ps 
->  ( ph  \/  ps ) )
75, 6impbid1 140 . . 3  |-  ( -. 
ph  ->  ( ( ph  \/  ps )  <->  ps )
)
84, 7syl 14 . 2  |-  ( -. 
E. x ph  ->  ( ( ph  \/  ps ) 
<->  ps ) )
92, 8eubidh 1954 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 664   E.wex 1426   E!weu 1948
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-eu 1951
This theorem is referenced by:  reuun2  3282
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