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Theorem nic-dfim 1424
Description: Define implication in terms of 'nand'. Analogous to  ( ( ph  -/\  ( ps  -/\  ps ) )  <->  ( ph  ->  ps ) ). 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-dfim  |-  ( ( ( ph  -/\  ( ps  -/\  ps ) ) 
-/\  ( ph  ->  ps ) )  -/\  (
( ( ph  -/\  ( ps  -/\  ps ) ) 
-/\  ( ph  -/\  ( ps  -/\  ps ) ) )  -/\  ( ( ph  ->  ps )  -/\  ( ph  ->  ps )
) ) )

Proof of Theorem nic-dfim
StepHypRef Expression
1 nanim 1292 . . 3  |-  ( (
ph  ->  ps )  <->  ( ph  -/\  ( ps  -/\  ps )
) )
21bicomi 193 . 2  |-  ( (
ph  -/\  ( ps  -/\  ps ) )  <->  ( ph  ->  ps ) )
3 nanbi 1294 . 2  |-  ( ( ( ph  -/\  ( ps  -/\  ps ) )  <-> 
( ph  ->  ps )
)  <->  ( ( (
ph  -/\  ( ps  -/\  ps ) )  -/\  ( ph  ->  ps ) ) 
-/\  ( ( (
ph  -/\  ( ps  -/\  ps ) )  -/\  ( ph  -/\  ( ps  -/\  ps ) ) )  -/\  ( ( ph  ->  ps )  -/\  ( ph  ->  ps ) ) ) ) )
42, 3mpbi 199 1  |-  ( ( ( ph  -/\  ( ps  -/\  ps ) ) 
-/\  ( ph  ->  ps ) )  -/\  (
( ( ph  -/\  ( ps  -/\  ps ) ) 
-/\  ( ph  -/\  ( ps  -/\  ps ) ) )  -/\  ( ( ph  ->  ps )  -/\  ( ph  ->  ps )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    -/\ wnan 1287
This theorem is referenced by:  nic-stdmp  1445  nic-luk1  1446  nic-luk2  1447  nic-luk3  1448
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-or 359  df-an 360  df-nan 1288
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