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Theorem reueq 2975
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 2603 . 2  |-  ( B  e.  A  <->  E. x  e.  A  x  =  B )
2 moeq 2954 . . . 4  |-  E* x  x  =  B
3 mormo 2765 . . . 4  |-  ( E* x  x  =  B  ->  E* x  e.  A x  =  B )
42, 3ax-mp 8 . . 3  |-  E* x  e.  A x  =  B
5 reu5 2766 . . 3  |-  ( E! x  e.  A  x  =  B  <->  ( E. x  e.  A  x  =  B  /\  E* x  e.  A x  =  B ) )
64, 5mpbiran2 885 . 2  |-  ( E! x  e.  A  x  =  B  <->  E. x  e.  A  x  =  B )
71, 6bitr4i 243 1  |-  ( B  e.  A  <->  E! x  e.  A  x  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   E*wmo 2157   E.wrex 2557   E!wreu 2558   E*wrmo 2559
This theorem is referenced by:  icoshftf1o  10775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-rex 2562  df-reu 2563  df-rmo 2564  df-v 2803
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