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Theorem euor2 2224
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 1718 . . 3  |-  F/ x E. x ph
21nfn 1777 . 2  |-  F/ x  -.  E. x ph
3 19.8a 1730 . . . 4  |-  ( ph  ->  E. x ph )
43con3i 127 . . 3  |-  ( -. 
E. x ph  ->  -. 
ph )
5 orel1 371 . . . 4  |-  ( -. 
ph  ->  ( ( ph  \/  ps )  ->  ps ) )
6 olc 373 . . . 4  |-  ( ps 
->  ( ph  \/  ps ) )
75, 6impbid1 194 . . 3  |-  ( -. 
ph  ->  ( ( ph  \/  ps )  <->  ps )
)
84, 7syl 15 . 2  |-  ( -. 
E. x ph  ->  ( ( ph  \/  ps ) 
<->  ps ) )
92, 8eubid 2163 1  |-  ( -. 
E. x ph  ->  ( E! x ( ph  \/  ps )  <->  E! x ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357   E.wex 1531   E!weu 2156
This theorem is referenced by:  reuun2  3464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-or 359  df-tru 1310  df-ex 1532  df-nf 1535  df-eu 2160
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