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Theorem reueq 2878
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 2461 . 2  |-  ( B  e.  A  <->  E. x  e.  A  x  =  B )
2 moeq 2854 . . . 4  |-  E* x  x  =  B
3 mormo 2640 . . . 4  |-  ( E* x  x  =  B  ->  E* x  e.  A  x  =  B )
42, 3ax-mp 5 . . 3  |-  E* x  e.  A  x  =  B
5 reu5 2641 . . 3  |-  ( E! x  e.  A  x  =  B  <->  ( E. x  e.  A  x  =  B  /\  E* x  e.  A  x  =  B ) )
64, 5mpbiran2 925 . 2  |-  ( E! x  e.  A  x  =  B  <->  E. x  e.  A  x  =  B )
71, 6bitr4i 186 1  |-  ( B  e.  A  <->  E! x  e.  A  x  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1331    e. wcel 1480   E*wmo 1998   E.wrex 2415   E!wreu 2416   E*wrmo 2417
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-rex 2420  df-reu 2421  df-rmo 2422  df-v 2683
This theorem is referenced by:  divfnzn  9406  icoshftf1o  9767
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