MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eueq Structured version   Unicode version

Theorem eueq 3268
Description: Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
eueq  |-  ( A  e.  _V  <->  E! x  x  =  A )
Distinct variable group:    x, A

Proof of Theorem eueq
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqtr3 2482 . . . 4  |-  ( ( x  =  A  /\  y  =  A )  ->  x  =  y )
21gen2 1624 . . 3  |-  A. x A. y ( ( x  =  A  /\  y  =  A )  ->  x  =  y )
32biantru 503 . 2  |-  ( E. x  x  =  A  <-> 
( E. x  x  =  A  /\  A. x A. y ( ( x  =  A  /\  y  =  A )  ->  x  =  y ) ) )
4 isset 3110 . 2  |-  ( A  e.  _V  <->  E. x  x  =  A )
5 eqeq1 2458 . . 3  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
65eu4 2336 . 2  |-  ( E! x  x  =  A  <-> 
( E. x  x  =  A  /\  A. x A. y ( ( x  =  A  /\  y  =  A )  ->  x  =  y ) ) )
73, 4, 63bitr4i 277 1  |-  ( A  e.  _V  <->  E! x  x  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    = wceq 1398   E.wex 1617    e. wcel 1823   E!weu 2284   _Vcvv 3106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3108
This theorem is referenced by:  eueq1  3269  moeq  3272  reuhypd  4664  mptfng  5688  upxp  20293  mptfnf  27724  iotasbc  31570
  Copyright terms: Public domain W3C validator