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Theorem eueq 3098
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 2454 . . . 4  |-  ( ( x  =  A  /\  y  =  A )  ->  x  =  y )
21gen2 1556 . . 3  |-  A. x A. y ( ( x  =  A  /\  y  =  A )  ->  x  =  y )
32biantru 492 . 2  |-  ( E. x  x  =  A  <-> 
( E. x  x  =  A  /\  A. x A. y ( ( x  =  A  /\  y  =  A )  ->  x  =  y ) ) )
4 isset 2952 . 2  |-  ( A  e.  _V  <->  E. x  x  =  A )
5 eqeq1 2441 . . 3  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
65eu4 2319 . 2  |-  ( E! x  x  =  A  <-> 
( E. x  x  =  A  /\  A. x A. y ( ( x  =  A  /\  y  =  A )  ->  x  =  y ) ) )
73, 4, 63bitr4i 269 1  |-  ( A  e.  _V  <->  E! x  x  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2280   _Vcvv 2948
This theorem is referenced by:  eueq1  3099  moeq  3102  reuhypd  4741  mptfng  5561  upxp  17643  mptfnf  24061  iotasbc  27534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-v 2950
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