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

Theorem euequ1 2316
Description: Equality has existential uniqueness. Special case of eueq1 3223 proved using only predicate calculus. The proof needs  y  =  z be free of  x. This is ensured by having  x and  y be distinct. Alternatively, a distinctor 
-.  A. x x  =  y could have been used instead. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof shortened by Wolf Lammen, 8-Sep-2019.)
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 ax6ev 1818 . . 3  |-  E. z 
z  =  y
2 equequ2 1879 . . . . 5  |-  ( y  =  z  ->  (
x  =  y  <->  x  =  z ) )
32equcoms 1875 . . . 4  |-  ( z  =  y  ->  (
x  =  y  <->  x  =  z ) )
43alrimiv 1784 . . 3  |-  ( z  =  y  ->  A. x
( x  =  y  <-> 
x  =  z ) )
51, 4eximii 1720 . 2  |-  E. z A. x ( x  =  y  <->  x  =  z
)
6 df-eu 2314 . 2  |-  ( E! x  x  =  y  <->  E. z A. x ( x  =  y  <->  x  =  z ) )
75, 6mpbir 214 1  |-  E! x  x  =  y
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   A.wal 1453   E.wex 1674   E!weu 2310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-eu 2314
This theorem is referenced by:  copsexg  4704  oprabid  6347
  Copyright terms: Public domain W3C validator