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Theorem eueq1 3271
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 3270 . 2  |-  ( A  e.  _V  <->  E! x  x  =  A )
31, 2mpbi 208 1  |-  E! x  x  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    e. wcel 1762   E!weu 2270   _Vcvv 3108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-v 3110
This theorem is referenced by:  eueq2  3272  eueq3  3273  fsn  6052  bj-nuliota  33544
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