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Theorem exists2 2072
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 1985 . . . . . 6  |-  ( E! x  x  =  x  ->  A. x E! x  x  =  x )
2 hba1 1503 . . . . . 6  |-  ( A. x ph  ->  A. x A. x ph )
3 exists1 2071 . . . . . . 7  |-  ( E! x  x  =  x  <->  A. x  x  =  y )
4 ax16 1767 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
53, 4sylbi 120 . . . . . 6  |-  ( E! x  x  =  x  ->  ( ph  ->  A. x ph ) )
61, 2, 5exlimdh 1558 . . . . 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 1607 . . . 4  |-  ( A. x ph  ->  -.  E. x  -.  ph )
97, 8syl6 33 . . 3  |-  ( E. x ph  ->  ( E! x  x  =  x  ->  -.  E. x  -.  ph ) )
109con2d 596 . 2  |-  ( E. x ph  ->  ( E. x  -.  ph  ->  -.  E! x  x  =  x ) )
1110imp 123 1  |-  ( ( E. x ph  /\  E. x  -.  ph )  ->  -.  E! x  x  =  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103   A.wal 1312    = wceq 1314   E.wex 1451   E!weu 1975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978
This theorem is referenced by: (None)
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