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Theorem euequ1 2371
Description: Equality has existential uniqueness. Special case of eueq1 3109 proved using only predicate calculus. (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 a9ev 1669 . 2  |-  E. x  x  =  y
2 equtr2 1701 . . 3  |-  ( ( x  =  y  /\  z  =  y )  ->  x  =  z )
32gen2 1557 . 2  |-  A. x A. z ( ( x  =  y  /\  z  =  y )  ->  x  =  z )
4 equequ1 1697 . . 3  |-  ( x  =  z  ->  (
x  =  y  <->  z  =  y ) )
54eu4 2322 . 2  |-  ( E! x  x  =  y  <-> 
( E. x  x  =  y  /\  A. x A. z ( ( x  =  y  /\  z  =  y )  ->  x  =  z ) ) )
61, 3, 5mpbir2an 888 1  |-  E! x  x  =  y
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   A.wal 1550   E.wex 1551   E!weu 2283
This theorem is referenced by:  copsexg  4444  oprabid  6107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288
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