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Theorem nic-dfim 1522
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 1352 . . 3  |-  ( (
ph  ->  ps )  <->  ( ph  -/\  ( ps  -/\  ps )
) )
21bicomi 202 . 2  |-  ( (
ph  -/\  ( ps  -/\  ps ) )  <->  ( ph  ->  ps ) )
3 nanbi 1354 . 2  |-  ( ( ( ph  -/\  ( ps  -/\  ps ) )  <-> 
( ph  ->  ps )
)  <->  ( ( (
ph  -/\  ( ps  -/\  ps ) )  -/\  ( ph  ->  ps ) ) 
-/\  ( ( (
ph  -/\  ( ps  -/\  ps ) )  -/\  ( ph  -/\  ( ps  -/\  ps ) ) )  -/\  ( ( ph  ->  ps )  -/\  ( ph  ->  ps ) ) ) ) )
42, 3mpbi 208 1  |-  ( ( ( ph  -/\  ( ps  -/\  ps ) ) 
-/\  ( ph  ->  ps ) )  -/\  (
( ( ph  -/\  ( ps  -/\  ps ) ) 
-/\  ( ph  -/\  ( ps  -/\  ps ) ) )  -/\  ( ( ph  ->  ps )  -/\  ( ph  ->  ps )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    -/\ wnan 1345
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-or 368  df-an 369  df-nan 1346
This theorem is referenced by:  nic-stdmp  1543  nic-luk1  1544  nic-luk2  1545  nic-luk3  1546
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