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Theorem euequ1 1995
Description: Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.)
Assertion
Ref Expression
euequ1  |-  E! x  x  =  y
Distinct variable group:    x, y

Proof of Theorem euequ1
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 a9e 1586 . 2  |-  E. x  x  =  y
2 equtr2 1597 . . 3  |-  ( ( x  =  y  /\  z  =  y )  ->  x  =  z )
32gen2 1339 . 2  |-  A. x A. z ( ( x  =  y  /\  z  =  y )  ->  x  =  z )
4 equequ1 1598 . . 3  |-  ( x  =  z  ->  (
x  =  y  <->  z  =  y ) )
54eu4 1962 . 2  |-  ( E! x  x  =  y  <-> 
( E. x  x  =  y  /\  A. x A. z ( ( x  =  y  /\  z  =  y )  ->  x  =  z ) ) )
61, 3, 5mpbir2an 849 1  |-  E! x  x  =  y
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97   A.wal 1241   E.wex 1381   E!weu 1900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904
This theorem is referenced by:  copsexg  3981  oprabid  5537
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