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Theorem eueq 3198
Description: Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
eueq  |-  ( A  e.  _V  <->  E! x  x  =  A )
Distinct variable group:    x, A

Proof of Theorem eueq
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqtr3 2492 . . . 4  |-  ( ( x  =  A  /\  y  =  A )  ->  x  =  y )
21gen2 1678 . . 3  |-  A. x A. y ( ( x  =  A  /\  y  =  A )  ->  x  =  y )
32biantru 513 . 2  |-  ( E. x  x  =  A  <-> 
( E. x  x  =  A  /\  A. x A. y ( ( x  =  A  /\  y  =  A )  ->  x  =  y ) ) )
4 isset 3035 . 2  |-  ( A  e.  _V  <->  E. x  x  =  A )
5 eqeq1 2475 . . 3  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
65eu4 2367 . 2  |-  ( E! x  x  =  A  <-> 
( E. x  x  =  A  /\  A. x A. y ( ( x  =  A  /\  y  =  A )  ->  x  =  y ) ) )
73, 4, 63bitr4i 285 1  |-  ( A  e.  _V  <->  E! x  x  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450    = wceq 1452   E.wex 1671    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:  eueq1  3199  moeq  3202  reuhypd  4627  mptfnf  5709  mptfng  5713  upxp  20715  iotasbc  36840
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