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Theorem exists2 2372
Description: A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
exists2  |-  ( ( E. x ph  /\  E. x  -.  ph )  ->  -.  E! x  x  =  x )

Proof of Theorem exists2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfeu1 2292 . . . . . 6  |-  F/ x E! x  x  =  x
2 nfa1 1807 . . . . . 6  |-  F/ x A. x ph
3 exists1 2371 . . . . . . 7  |-  ( E! x  x  =  x  <->  A. x  x  =  y )
4 ax16 2051 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
53, 4sylbi 189 . . . . . 6  |-  ( E! x  x  =  x  ->  ( ph  ->  A. x ph ) )
61, 2, 5exlimd 1825 . . . . 5  |-  ( E! x  x  =  x  ->  ( E. x ph  ->  A. x ph )
)
76com12 30 . . . 4  |-  ( E. x ph  ->  ( E! x  x  =  x  ->  A. x ph )
)
8 alex 1582 . . . 4  |-  ( A. x ph  <->  -.  E. x  -.  ph )
97, 8syl6ib 219 . . 3  |-  ( E. x ph  ->  ( E! x  x  =  x  ->  -.  E. x  -.  ph ) )
109con2d 110 . 2  |-  ( E. x ph  ->  ( E. x  -.  ph  ->  -.  E! x  x  =  x ) )
1110imp 420 1  |-  ( ( E. x ph  /\  E. x  -.  ph )  ->  -.  E! x  x  =  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   A.wal 1550   E.wex 1551   E!weu 2282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555  df-eu 2286
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