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Theorem euor2 2307
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 nfe1 1739 . . 3  |-  F/ x E. x ph
21nfn 1801 . 2  |-  F/ x  -.  E. x ph
3 19.8a 1754 . . . 4  |-  ( ph  ->  E. x ph )
43con3i 129 . . 3  |-  ( -. 
E. x ph  ->  -. 
ph )
5 orel1 372 . . . 4  |-  ( -. 
ph  ->  ( ( ph  \/  ps )  ->  ps ) )
6 olc 374 . . . 4  |-  ( ps 
->  ( ph  \/  ps ) )
75, 6impbid1 195 . . 3  |-  ( -. 
ph  ->  ( ( ph  \/  ps )  <->  ps )
)
84, 7syl 16 . 2  |-  ( -. 
E. x ph  ->  ( ( ph  \/  ps ) 
<->  ps ) )
92, 8eubid 2246 1  |-  ( -. 
E. x ph  ->  ( E! x ( ph  \/  ps )  <->  E! x ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358   E.wex 1547   E!weu 2239
This theorem is referenced by:  reuun2  3568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-or 360  df-tru 1325  df-ex 1548  df-nf 1551  df-eu 2243
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