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Theorem eueq1 3099
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 3098 . 2  |-  ( A  e.  _V  <->  E! x  x  =  A )
31, 2mpbi 200 1  |-  E! x  x  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   E!weu 2280   _Vcvv 2948
This theorem is referenced by:  eueq2  3100  eueq3  3101  fsn  5898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-v 2950
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