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Theorem reueq 3258
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 2879 . 2  |-  ( B  e.  A  <->  E. x  e.  A  x  =  B )
2 moeq 3236 . . . 4  |-  E* x  x  =  B
3 mormo 3035 . . . 4  |-  ( E* x  x  =  B  ->  E* x  e.  A  x  =  B )
42, 3ax-mp 5 . . 3  |-  E* x  e.  A  x  =  B
5 reu5 3036 . . 3  |-  ( E! x  e.  A  x  =  B  <->  ( E. x  e.  A  x  =  B  /\  E* x  e.  A  x  =  B ) )
64, 5mpbiran2 910 . 2  |-  ( E! x  e.  A  x  =  B  <->  E. x  e.  A  x  =  B )
71, 6bitr4i 252 1  |-  ( B  e.  A  <->  E! x  e.  A  x  =  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758   E*wmo 2262   E.wrex 2797   E!wreu 2798   E*wrmo 2799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-rex 2802  df-reu 2803  df-rmo 2804  df-v 3074
This theorem is referenced by:  icoshftf1o  11520
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