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Theorem exists1 2369
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; see theorem dtru 4382. (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 2284 . 2  |-  ( E! x  x  =  x  <->  E. y A. x ( x  =  x  <->  x  =  y ) )
2 equid 1688 . . . . . 6  |-  x  =  x
32tbt 334 . . . . 5  |-  ( x  =  y  <->  ( x  =  y  <->  x  =  x
) )
4 bicom 192 . . . . 5  |-  ( ( x  =  y  <->  x  =  x )  <->  ( x  =  x  <->  x  =  y
) )
53, 4bitri 241 . . . 4  |-  ( x  =  y  <->  ( x  =  x  <->  x  =  y
) )
65albii 1575 . . 3  |-  ( A. x  x  =  y  <->  A. x ( x  =  x  <->  x  =  y
) )
76exbii 1592 . 2  |-  ( E. y A. x  x  =  y  <->  E. y A. x ( x  =  x  <->  x  =  y
) )
8 nfae 2042 . . 3  |-  F/ y A. x  x  =  y
9819.9 1797 . 2  |-  ( E. y A. x  x  =  y  <->  A. x  x  =  y )
101, 7, 93bitr2i 265 1  |-  ( E! x  x  =  x  <->  A. x  x  =  y )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1549   E.wex 1550   E!weu 2280
This theorem is referenced by:  exists2  2370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-eu 2284
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