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Theorem eueq 2937
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 2302 . . . 4  |-  ( ( x  =  A  /\  y  =  A )  ->  x  =  y )
21gen2 1534 . . 3  |-  A. x A. y ( ( x  =  A  /\  y  =  A )  ->  x  =  y )
32biantru 491 . 2  |-  ( E. x  x  =  A  <-> 
( E. x  x  =  A  /\  A. x A. y ( ( x  =  A  /\  y  =  A )  ->  x  =  y ) ) )
4 isset 2792 . 2  |-  ( A  e.  _V  <->  E. x  x  =  A )
5 eqeq1 2289 . . 3  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
65eu4 2182 . 2  |-  ( E! x  x  =  A  <-> 
( E. x  x  =  A  /\  A. x A. y ( ( x  =  A  /\  y  =  A )  ->  x  =  y ) ) )
73, 4, 63bitr4i 268 1  |-  ( A  e.  _V  <->  E! x  x  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   _Vcvv 2788
This theorem is referenced by:  eueq1  2938  moeq  2941  reuhypd  4561  mptfng  5369  upxp  17317  mptfnf  23226  iotasbc  27619
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
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