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Theorem nandsym1 24933
Description: A symmetry with  -/\.

See negsym1 24928 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

Assertion
Ref Expression
nandsym1  |-  ( ( ps  -/\  ( ps  -/\ 
F.  ) )  -> 
( ps  -/\  ph )
)

Proof of Theorem nandsym1
StepHypRef Expression
1 df-nan 1288 . . . . 5  |-  ( ( ps  -/\  ( ps  -/\ 
F.  ) )  <->  -.  ( ps  /\  ( ps  -/\  F.  ) ) )
21biimpi 186 . . . 4  |-  ( ( ps  -/\  ( ps  -/\ 
F.  ) )  ->  -.  ( ps  /\  ( ps  -/\  F.  ) ) )
3 df-nan 1288 . . . . 5  |-  ( ( ps  -/\  F.  )  <->  -.  ( ps  /\  F.  ) )
43anbi2i 675 . . . 4  |-  ( ( ps  /\  ( ps 
-/\  F.  ) )  <->  ( ps  /\  -.  ( ps  /\  F.  ) ) )
52, 4sylnib 295 . . 3  |-  ( ( ps  -/\  ( ps  -/\ 
F.  ) )  ->  -.  ( ps  /\  -.  ( ps  /\  F.  )
) )
6 simpl 443 . . . 4  |-  ( ( ps  /\  ph )  ->  ps )
7 fal 1313 . . . . 5  |-  -.  F.
87intnan 880 . . . 4  |-  -.  ( ps  /\  F.  )
96, 8jctir 524 . . 3  |-  ( ( ps  /\  ph )  ->  ( ps  /\  -.  ( ps  /\  F.  )
) )
105, 9nsyl 113 . 2  |-  ( ( ps  -/\  ( ps  -/\ 
F.  ) )  ->  -.  ( ps  /\  ph ) )
11 df-nan 1288 . 2  |-  ( ( ps  -/\  ph )  <->  -.  ( ps  /\  ph ) )
1210, 11sylibr 203 1  |-  ( ( ps  -/\  ( ps  -/\ 
F.  ) )  -> 
( ps  -/\  ph )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    -/\ wnan 1287    F. wfal 1308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360  df-nan 1288  df-tru 1310  df-fal 1311
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