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Theorem eueq1 2773
Description: Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)
Hypothesis
Ref Expression
eueq1.1  |-  A  e. 
_V
Assertion
Ref Expression
eueq1  |-  E! x  x  =  A
Distinct variable group:    x, A

Proof of Theorem eueq1
StepHypRef Expression
1 eueq1.1 . 2  |-  A  e. 
_V
2 eueq 2772 . 2  |-  ( A  e.  _V  <->  E! x  x  =  A )
31, 2mpbi 143 1  |-  E! x  x  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434   E!weu 1943   _Vcvv 2610
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  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-v 2612
This theorem is referenced by:  eueq2dc  2774  eueq3dc  2775  fsn  5388
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