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Theorem exists2 2013
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 1926 . . . . . 6  |-  ( E! x  x  =  x  ->  A. x E! x  x  =  x )
2 hba1 1449 . . . . . 6  |-  ( A. x ph  ->  A. x A. x ph )
3 exists1 2012 . . . . . . 7  |-  ( E! x  x  =  x  <->  A. x  x  =  y )
4 ax16 1710 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
53, 4sylbi 118 . . . . . 6  |-  ( E! x  x  =  x  ->  ( ph  ->  A. x ph ) )
61, 2, 5exlimdh 1503 . . . . 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 1552 . . . 4  |-  ( A. x ph  ->  -.  E. x  -.  ph )
97, 8syl6 33 . . 3  |-  ( E. x ph  ->  ( E! x  x  =  x  ->  -.  E. x  -.  ph ) )
109con2d 564 . 2  |-  ( E. x ph  ->  ( E. x  -.  ph  ->  -.  E! x  x  =  x ) )
1110imp 119 1  |-  ( ( E. x ph  /\  E. x  -.  ph )  ->  -.  E! x  x  =  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101   A.wal 1257    = wceq 1259   E.wex 1397   E!weu 1916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919
This theorem is referenced by: (None)
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