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Theorem amosym1 31085
Description: A symmetry with  E*.

See negsym1 31076 for more information. (Contributed by Anthony Hart, 13-Sep-2011.)

Assertion
Ref Expression
amosym1  |-  ( E* x E* x F.  ->  E* x ph )

Proof of Theorem amosym1
StepHypRef Expression
1 df-mo 2271 . 2  |-  ( E* x E* x F.  <->  ( E. x E* x F.  ->  E! x E* x F.  ) )
2 mof 31069 . . . . 5  |-  E* x F.
3 19.8a 1909 . . . . . . 7  |-  ( E* x F.  ->  E. x E* x F.  )
43pm2.24d 138 . . . . . 6  |-  ( E* x F.  ->  ( -.  E. x E* x F.  ->  -.  -.  E. x E* x F.  )
)
54pm2.01d 173 . . . . 5  |-  ( E* x F.  ->  -.  -.  E. x E* x F.  )
62, 5ax-mp 5 . . . 4  |-  -.  -.  E. x E* x F.
76pm2.21i 135 . . 3  |-  ( -. 
E. x E* x F.  ->  E* x ph )
82notnoti 127 . . . . . 6  |-  -.  -.  E* x F.
98nex 1675 . . . . 5  |-  -.  E. x  -.  E* x F.
10 eunex 4615 . . . . 5  |-  ( E! x E* x F.  ->  E. x  -.  E* x F.  )
119, 10mto 180 . . . 4  |-  -.  E! x E* x F.
1211pm2.21i 135 . . 3  |-  ( E! x E* x F.  ->  E* x ph )
137, 12ja 165 . 2  |-  ( ( E. x E* x F.  ->  E! x E* x F.  )  ->  E* x ph )
141, 13sylbi 199 1  |-  ( E* x E* x F.  ->  E* x ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   F. wfal 1443   E.wex 1660   E!weu 2266   E*wmo 2267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-nul 4553  ax-pow 4600
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-eu 2270  df-mo 2271
This theorem is referenced by: (None)
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