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Theorem reueq 3131
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 2753 . 2  |-  ( B  e.  A  <->  E. x  e.  A  x  =  B )
2 moeq 3110 . . . 4  |-  E* x  x  =  B
3 mormo 2920 . . . 4  |-  ( E* x  x  =  B  ->  E* x  e.  A x  =  B )
42, 3ax-mp 8 . . 3  |-  E* x  e.  A x  =  B
5 reu5 2921 . . 3  |-  ( E! x  e.  A  x  =  B  <->  ( E. x  e.  A  x  =  B  /\  E* x  e.  A x  =  B ) )
64, 5mpbiran2 886 . 2  |-  ( E! x  e.  A  x  =  B  <->  E. x  e.  A  x  =  B )
71, 6bitr4i 244 1  |-  ( B  e.  A  <->  E! x  e.  A  x  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   E*wmo 2282   E.wrex 2706   E!wreu 2707   E*wrmo 2708
This theorem is referenced by:  icoshftf1o  11020
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 2417
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-rex 2711  df-reu 2712  df-rmo 2713  df-v 2958
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