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Theorem euequ1 2038
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 1627 . 2  |-  E. x  x  =  y
2 equtr2 1639 . . 3  |-  ( ( x  =  y  /\  z  =  y )  ->  x  =  z )
32gen2 1380 . 2  |-  A. x A. z ( ( x  =  y  /\  z  =  y )  ->  x  =  z )
4 equequ1 1640 . . 3  |-  ( x  =  z  ->  (
x  =  y  <->  z  =  y ) )
54eu4 2005 . 2  |-  ( E! x  x  =  y  <-> 
( E. x  x  =  y  /\  A. x A. z ( ( x  =  y  /\  z  =  y )  ->  x  =  z ) ) )
61, 3, 5mpbir2an 884 1  |-  E! x  x  =  y
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1283   E.wex 1422   E!weu 1943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947
This theorem is referenced by:  copsexg  4027  oprabid  5588
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