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Theorem euequ1 2070
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 1657 . 2  |-  E. x  x  =  y
2 equtr2 1670 . . 3  |-  ( ( x  =  y  /\  z  =  y )  ->  x  =  z )
32gen2 1409 . 2  |-  A. x A. z ( ( x  =  y  /\  z  =  y )  ->  x  =  z )
4 equequ1 1671 . . 3  |-  ( x  =  z  ->  (
x  =  y  <->  z  =  y ) )
54eu4 2037 . 2  |-  ( E! x  x  =  y  <-> 
( E. x  x  =  y  /\  A. x A. z ( ( x  =  y  /\  z  =  y )  ->  x  =  z ) ) )
61, 3, 5mpbir2an 909 1  |-  E! x  x  =  y
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1312   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-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  df-mo 1979
This theorem is referenced by:  copsexg  4134  oprabid  5769
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