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Theorem reueq 3301
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 2987 . 2  |-  ( B  e.  A  <->  E. x  e.  A  x  =  B )
2 moeq 3279 . . . 4  |-  E* x  x  =  B
3 mormo 3076 . . . 4  |-  ( E* x  x  =  B  ->  E* x  e.  A  x  =  B )
42, 3ax-mp 5 . . 3  |-  E* x  e.  A  x  =  B
5 reu5 3077 . . 3  |-  ( E! x  e.  A  x  =  B  <->  ( E. x  e.  A  x  =  B  /\  E* x  e.  A  x  =  B ) )
64, 5mpbiran2 917 . 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 1379    e. wcel 1767   E*wmo 2276   E.wrex 2815   E!wreu 2816   E*wrmo 2817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-rex 2820  df-reu 2821  df-rmo 2822  df-v 3115
This theorem is referenced by:  icoshftf1o  11639
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