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Theorem exists2 2045
Description: A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
exists2  |-  ( ( E. x ph  /\  E. x  -.  ph )  ->  -.  E! x  x  =  x )

Proof of Theorem exists2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 hbeu1 1958 . . . . . 6  |-  ( E! x  x  =  x  ->  A. x E! x  x  =  x )
2 hba1 1478 . . . . . 6  |-  ( A. x ph  ->  A. x A. x ph )
3 exists1 2044 . . . . . . 7  |-  ( E! x  x  =  x  <->  A. x  x  =  y )
4 ax16 1741 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
53, 4sylbi 119 . . . . . 6  |-  ( E! x  x  =  x  ->  ( ph  ->  A. x ph ) )
61, 2, 5exlimdh 1532 . . . . 5  |-  ( E! x  x  =  x  ->  ( E. x ph  ->  A. x ph )
)
76com12 30 . . . 4  |-  ( E. x ph  ->  ( E! x  x  =  x  ->  A. x ph )
)
8 alexim 1581 . . . 4  |-  ( A. x ph  ->  -.  E. x  -.  ph )
97, 8syl6 33 . . 3  |-  ( E. x ph  ->  ( E! x  x  =  x  ->  -.  E. x  -.  ph ) )
109con2d 589 . 2  |-  ( E. x ph  ->  ( E. x  -.  ph  ->  -.  E! x  x  =  x ) )
1110imp 122 1  |-  ( ( E. x ph  /\  E. x  -.  ph )  ->  -.  E! x  x  =  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102   A.wal 1287    = wceq 1289   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-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951
This theorem is referenced by: (None)
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