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Theorem euequ1 2269
Description: Equality has existential uniqueness. Special case of eueq1 3241 proved using only predicate calculus. The proof needs  y  =  z be free of  x. This is ensured by having  x and  y be distinct. Alternatevly, 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 1796 . . 3  |-  E. z 
z  =  y
2 equequ2 1848 . . . . 5  |-  ( y  =  z  ->  (
x  =  y  <->  x  =  z ) )
32equcoms 1844 . . . 4  |-  ( z  =  y  ->  (
x  =  y  <->  x  =  z ) )
43alrimiv 1763 . . 3  |-  ( z  =  y  ->  A. x
( x  =  y  <-> 
x  =  z ) )
51, 4eximii 1704 . 2  |-  E. z A. x ( x  =  y  <->  x  =  z
)
6 df-eu 2267 . 2  |-  ( E! x  x  =  y  <->  E. z A. x ( x  =  y  <->  x  =  z ) )
75, 6mpbir 212 1  |-  E! x  x  =  y
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   A.wal 1435   E.wex 1659   E!weu 2263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838
This theorem depends on definitions:  df-bi 188  df-ex 1660  df-eu 2267
This theorem is referenced by:  copsexg  4699  oprabid  6324
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