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Theorem eueq1 3222
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 3221 . 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 1405    e. wcel 1842   E!weu 2238   _Vcvv 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-v 3061
This theorem is referenced by:  eueq2  3223  eueq3  3224  fsn  6048  bj-nuliota  31151
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