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Theorem eueq 2901
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 2190 . . . 4  |-  ( ( x  =  A  /\  y  =  A )  ->  x  =  y )
21gen2 1443 . . 3  |-  A. x A. y ( ( x  =  A  /\  y  =  A )  ->  x  =  y )
32biantru 300 . 2  |-  ( E. x  x  =  A  <-> 
( E. x  x  =  A  /\  A. x A. y ( ( x  =  A  /\  y  =  A )  ->  x  =  y ) ) )
4 isset 2736 . 2  |-  ( A  e.  _V  <->  E. x  x  =  A )
5 eqeq1 2177 . . 3  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
65eu4 2081 . 2  |-  ( E! x  x  =  A  <-> 
( E. x  x  =  A  /\  A. x A. y ( ( x  =  A  /\  y  =  A )  ->  x  =  y ) ) )
73, 4, 63bitr4i 211 1  |-  ( A  e.  _V  <->  E! x  x  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1346    = wceq 1348   E.wex 1485   E!weu 2019    e. wcel 2141   _Vcvv 2730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732
This theorem is referenced by:  eueq1  2902  moeq  2905  mosubt  2907  reuhypd  4456  mptfng  5323  upxp  13066
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