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Theorem nic-dfim 1429
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 1297 . . 3  |-  ( (
ph  ->  ps )  <->  ( ph  -/\  ( ps  -/\  ps )
) )
21bicomi 195 . 2  |-  ( (
ph  -/\  ( ps  -/\  ps ) )  <->  ( ph  ->  ps ) )
3 nanbi 1299 . 2  |-  ( ( ( ph  -/\  ( ps  -/\  ps ) )  <-> 
( ph  ->  ps )
)  <->  ( ( (
ph  -/\  ( ps  -/\  ps ) )  -/\  ( ph  ->  ps ) ) 
-/\  ( ( (
ph  -/\  ( ps  -/\  ps ) )  -/\  ( ph  -/\  ( ps  -/\  ps ) ) )  -/\  ( ( ph  ->  ps )  -/\  ( ph  ->  ps ) ) ) ) )
42, 3mpbi 201 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 6    <-> wb 178    -/\ wnan 1292
This theorem is referenced by:  nic-stdmp  1450  nic-luk1  1451  nic-luk2  1452  nic-luk3  1453
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-nan 1293
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