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Theorem euor2 2313
Description: Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
Assertion
Ref Expression
euor2  |-  ( -. 
E. x ph  ->  ( E! x ( ph  \/  ps )  <->  E! x ps ) )

Proof of Theorem euor2
StepHypRef Expression
1 nfe1 1894 . . 3  |-  F/ x E. x ph
21nfn 1960 . 2  |-  F/ x  -.  E. x ph
3 19.8a 1912 . . . 4  |-  ( ph  ->  E. x ph )
43con3i 140 . . 3  |-  ( -. 
E. x ph  ->  -. 
ph )
5 biorf 406 . . . 4  |-  ( -. 
ph  ->  ( ps  <->  ( ph  \/  ps ) ) )
65bicomd 204 . . 3  |-  ( -. 
ph  ->  ( ( ph  \/  ps )  <->  ps )
)
74, 6syl 17 . 2  |-  ( -. 
E. x ph  ->  ( ( ph  \/  ps ) 
<->  ps ) )
82, 7eubid 2287 1  |-  ( -. 
E. x ph  ->  ( E! x ( ph  \/  ps )  <->  E! x ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369   E.wex 1657   E!weu 2269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909
This theorem depends on definitions:  df-bi 188  df-or 371  df-ex 1658  df-nf 1662  df-eu 2273
This theorem is referenced by:  reuun2  3756
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