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Theorem nandsym1 28216
Description: A symmetry with  -/\.

See negsym1 28211 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 1334 . . . . 5  |-  ( ( ps  -/\  ( ps  -/\ F.  ) )  <->  -.  ( ps  /\  ( ps  -/\ F.  ) ) )
21biimpi 194 . . . 4  |-  ( ( ps  -/\  ( ps  -/\ F.  ) )  ->  -.  ( ps  /\  ( ps  -/\ F.  ) ) )
3 df-nan 1334 . . . . 5  |-  ( ( ps  -/\ F.  )  <->  -.  ( ps  /\ F.  ) )
43anbi2i 694 . . . 4  |-  ( ( ps  /\  ( ps 
-/\ F.  ) )  <->  ( ps  /\  -.  ( ps  /\ F.  ) ) )
52, 4sylnib 304 . . 3  |-  ( ( ps  -/\  ( ps  -/\ F.  ) )  ->  -.  ( ps  /\  -.  ( ps  /\ F.  ) ) )
6 simpl 457 . . . 4  |-  ( ( ps  /\  ph )  ->  ps )
7 fal 1376 . . . . 5  |-  -. F.
87intnan 905 . . . 4  |-  -.  ( ps  /\ F.  )
96, 8jctir 538 . . 3  |-  ( ( ps  /\  ph )  ->  ( ps  /\  -.  ( ps  /\ F.  )
) )
105, 9nsyl 121 . 2  |-  ( ( ps  -/\  ( ps  -/\ F.  ) )  ->  -.  ( ps  /\  ph )
)
11 df-nan 1334 . 2  |-  ( ( ps  -/\  ph )  <->  -.  ( ps  /\  ph ) )
1210, 11sylibr 212 1  |-  ( ( ps  -/\  ( ps  -/\ F.  ) )  ->  ( ps  -/\  ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    -/\ wnan 1333   F. wfal 1374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 371  df-nan 1334  df-tru 1372  df-fal 1375
This theorem is referenced by: (None)
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