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Theorem euex 2005
Description: Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
euex  |-  ( E! x ph  ->  E. x ph )

Proof of Theorem euex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ax-17 1489 . . 3  |-  ( ph  ->  A. y ph )
21eu1 2000 . 2  |-  ( E! x ph  <->  E. x
( ph  /\  A. y
( [ y  /  x ] ph  ->  x  =  y ) ) )
3 exsimpl 1579 . 2  |-  ( E. x ( ph  /\  A. y ( [ y  /  x ] ph  ->  x  =  y ) )  ->  E. x ph )
42, 3sylbi 120 1  |-  ( E! x ph  ->  E. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1312   E.wex 1451   [wsb 1718   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-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-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-eu 1978
This theorem is referenced by:  eu2  2019  eu3h  2020  eu5  2022  exmoeudc  2038  eupickbi  2057  2eu2ex  2064  euxfrdc  2841  repizf  4012  eusvnf  4342  eusvnfb  4343  tz6.12c  5417  ndmfvg  5418  nfvres  5420  0fv  5422  eusvobj2  5726  fnoprabg  5838  txcn  12339
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