MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  euequ1 Unicode version

Theorem euequ1 2231
Description: Equality has existential uniqueness. Special case of eueq1 2938 proved using only predicate calculus. (Contributed by Stefan Allan, 4-Dec-2008.)
Assertion
Ref Expression
euequ1  |-  E! x  x  =  y
Distinct variable group:    x, y

Proof of Theorem euequ1
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 a9ev 1637 . 2  |-  E. x  x  =  y
2 equtr2 1654 . . 3  |-  ( ( x  =  y  /\  z  =  y )  ->  x  =  z )
32gen2 1534 . 2  |-  A. x A. z ( ( x  =  y  /\  z  =  y )  ->  x  =  z )
4 equequ1 1648 . . 3  |-  ( x  =  z  ->  (
x  =  y  <->  z  =  y ) )
54eu4 2182 . 2  |-  ( E! x  x  =  y  <-> 
( E. x  x  =  y  /\  A. x A. z ( ( x  =  y  /\  z  =  y )  ->  x  =  z ) ) )
61, 3, 5mpbir2an 886 1  |-  E! x  x  =  y
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528   E!weu 2143
This theorem is referenced by:  copsexg  4252  oprabid  5882  copsexgb  24966
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148
  Copyright terms: Public domain W3C validator