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Theorem exists1 2044
Description: Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
exists1  |-  ( E! x  x  =  x  <->  A. x  x  =  y )
Distinct variable group:    x, y

Proof of Theorem exists1
StepHypRef Expression
1 df-eu 1951 . 2  |-  ( E! x  x  =  x  <->  E. y A. x ( x  =  x  <->  x  =  y ) )
2 equid 1634 . . . . . 6  |-  x  =  x
32tbt 245 . . . . 5  |-  ( x  =  y  <->  ( x  =  y  <->  x  =  x
) )
4 bicom 138 . . . . 5  |-  ( ( x  =  y  <->  x  =  x )  <->  ( x  =  x  <->  x  =  y
) )
53, 4bitri 182 . . . 4  |-  ( x  =  y  <->  ( x  =  x  <->  x  =  y
) )
65albii 1404 . . 3  |-  ( A. x  x  =  y  <->  A. x ( x  =  x  <->  x  =  y
) )
76exbii 1541 . 2  |-  ( E. y A. x  x  =  y  <->  E. y A. x ( x  =  x  <->  x  =  y
) )
8 hbae 1653 . . 3  |-  ( A. x  x  =  y  ->  A. y A. x  x  =  y )
9819.9h 1579 . 2  |-  ( E. y A. x  x  =  y  <->  A. x  x  =  y )
101, 7, 93bitr2i 206 1  |-  ( E! x  x  =  x  <->  A. x  x  =  y )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1287   E.wex 1426   E!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-eu 1951
This theorem is referenced by:  exists2  2045
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