MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eueq1 Unicode version

Theorem eueq1 2938
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 2937 . 2  |-  ( A  e.  _V  <->  E! x  x  =  A )
31, 2mpbi 199 1  |-  E! x  x  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   E!weu 2143   _Vcvv 2788
This theorem is referenced by:  eueq2  2939  eueq3  2940  fsn  5696
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790
  Copyright terms: Public domain W3C validator