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Theorem reueq 3275
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 2960 . 2  |-  ( B  e.  A  <->  E. x  e.  A  x  =  B )
2 moeq 3253 . . . 4  |-  E* x  x  =  B
3 mormo 3050 . . . 4  |-  ( E* x  x  =  B  ->  E* x  e.  A  x  =  B )
42, 3ax-mp 5 . . 3  |-  E* x  e.  A  x  =  B
5 reu5 3051 . . 3  |-  ( E! x  e.  A  x  =  B  <->  ( E. x  e.  A  x  =  B  /\  E* x  e.  A  x  =  B ) )
64, 5mpbiran2 927 . 2  |-  ( E! x  e.  A  x  =  B  <->  E. x  e.  A  x  =  B )
71, 6bitr4i 255 1  |-  ( B  e.  A  <->  E! x  e.  A  x  =  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437    e. wcel 1870   E*wmo 2267   E.wrex 2783   E!wreu 2784   E*wrmo 2785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-rex 2788  df-reu 2789  df-rmo 2790  df-v 3089
This theorem is referenced by:  icoshftf1o  11753
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