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Theorem eueq1 3199
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 3198 . 2  |-  ( A  e.  _V  <->  E! x  x  =  A )
31, 2mpbi 213 1  |-  E! x  x  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452    e. wcel 1904   E!weu 2319   _Vcvv 3031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-v 3033
This theorem is referenced by:  eueq2  3200  eueq3  3201  fsn  6077  bj-nuliota  31693
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