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Theorem reueq 2790
Description: Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
Assertion
Ref Expression
reueq  |-  ( B  e.  A  <->  E! x  e.  A  x  =  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem reueq
StepHypRef Expression
1 risset 2395 . 2  |-  ( B  e.  A  <->  E. x  e.  A  x  =  B )
2 moeq 2768 . . . 4  |-  E* x  x  =  B
3 mormo 2566 . . . 4  |-  ( E* x  x  =  B  ->  E* x  e.  A  x  =  B )
42, 3ax-mp 7 . . 3  |-  E* x  e.  A  x  =  B
5 reu5 2567 . . 3  |-  ( E! x  e.  A  x  =  B  <->  ( E. x  e.  A  x  =  B  /\  E* x  e.  A  x  =  B ) )
64, 5mpbiran2 883 . 2  |-  ( E! x  e.  A  x  =  B  <->  E. x  e.  A  x  =  B )
71, 6bitr4i 185 1  |-  ( B  e.  A  <->  E! x  e.  A  x  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1285    e. wcel 1434   E*wmo 1943   E.wrex 2350   E!wreu 2351   E*wrmo 2352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-rex 2355  df-reu 2356  df-rmo 2357  df-v 2604
This theorem is referenced by:  divfnzn  8776  icoshftf1o  9078
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