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Theorem nic-dfneg 1444
Description: Define negation in terms of 'nand'. Analogous to  ( ( ph  -/\  ph )  <->  -.  ph ). In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition ($a statement). (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nic-dfneg  |-  ( ( ( ph  -/\  ph )  -/\  -.  ph )  -/\  ( ( ( ph  -/\  ph )  -/\  ( ph  -/\  ph ) )  -/\  ( -.  ph  -/\  -.  ph )
) )

Proof of Theorem nic-dfneg
StepHypRef Expression
1 nannot 1302 . . 3  |-  ( -. 
ph 
<->  ( ph  -/\  ph )
)
21bicomi 194 . 2  |-  ( (
ph  -/\  ph )  <->  -.  ph )
3 nanbi 1303 . 2  |-  ( ( ( ph  -/\  ph )  <->  -. 
ph )  <->  ( (
( ph  -/\  ph )  -/\  -.  ph )  -/\  ( ( ( ph  -/\  ph )  -/\  ( ph  -/\  ph ) )  -/\  ( -.  ph  -/\  -.  ph )
) ) )
42, 3mpbi 200 1  |-  ( ( ( ph  -/\  ph )  -/\  -.  ph )  -/\  ( ( ( ph  -/\  ph )  -/\  ( ph  -/\  ph ) )  -/\  ( -.  ph  -/\  -.  ph )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    -/\ wnan 1296
This theorem is referenced by:  nic-luk2  1466  nic-luk3  1467
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 178  df-or 360  df-an 361  df-nan 1297
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