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Theorem amosym1 30044
Description: A symmetry with  E*.

See negsym1 30035 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 2223 . 2  |-  ( E* x E* x F.  <->  ( E. x E* x F.  ->  E! x E* x F.  ) )
2 mof 30028 . . . . 5  |-  E* x F.
3 19.8a 1865 . . . . . . 7  |-  ( E* x F.  ->  E. x E* x F.  )
43pm2.24d 143 . . . . . 6  |-  ( E* x F.  ->  ( -.  E. x E* x F.  ->  -.  -.  E. x E* x F.  )
)
54pm2.01d 169 . . . . 5  |-  ( E* x F.  ->  -.  -.  E. x E* x F.  )
62, 5ax-mp 5 . . . 4  |-  -.  -.  E. x E* x F.
76pm2.21i 131 . . 3  |-  ( -. 
E. x E* x F.  ->  E* x ph )
82notnoti 123 . . . . . 6  |-  -.  -.  E* x F.
98nex 1635 . . . . 5  |-  -.  E. x  -.  E* x F.
10 eunex 4558 . . . . 5  |-  ( E! x E* x F.  ->  E. x  -.  E* x F.  )
119, 10mto 176 . . . 4  |-  -.  E! x E* x F.
1211pm2.21i 131 . . 3  |-  ( E! x E* x F.  ->  E* x ph )
137, 12ja 161 . 2  |-  ( ( E. x E* x F.  ->  E! x E* x F.  )  ->  E* x ph )
141, 13sylbi 195 1  |-  ( E* x E* x F.  ->  E* x ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   F. wfal 1404   E.wex 1620   E!weu 2218   E*wmo 2219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-nul 4496  ax-pow 4543
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-eu 2222  df-mo 2223
This theorem is referenced by: (None)
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