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Theorem exists1 2232
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 4201. (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 2147 . 2  |-  ( E! x  x  =  x  <->  E. y A. x ( x  =  x  <->  x  =  y ) )
2 equid 1644 . . . . . 6  |-  x  =  x
32tbt 333 . . . . 5  |-  ( x  =  y  <->  ( x  =  y  <->  x  =  x
) )
4 bicom 191 . . . . 5  |-  ( ( x  =  y  <->  x  =  x )  <->  ( x  =  x  <->  x  =  y
) )
53, 4bitri 240 . . . 4  |-  ( x  =  y  <->  ( x  =  x  <->  x  =  y
) )
65albii 1553 . . 3  |-  ( A. x  x  =  y  <->  A. x ( x  =  x  <->  x  =  y
) )
76exbii 1569 . 2  |-  ( E. y A. x  x  =  y  <->  E. y A. x ( x  =  x  <->  x  =  y
) )
8 nfae 1894 . . 3  |-  F/ y A. x  x  =  y
9819.9 1783 . 2  |-  ( E. y A. x  x  =  y  <->  A. x  x  =  y )
101, 7, 93bitr2i 264 1  |-  ( E! x  x  =  x  <->  A. x  x  =  y )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1527   E.wex 1528    = wceq 1623   E!weu 2143
This theorem is referenced by:  exists2  2233
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-an 360  df-ex 1529  df-nf 1532  df-eu 2147
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