MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  renicax Unicode version

Theorem renicax 1452
Description: A rederivation of nic-ax 1428 from lukshef-ax1 1449, proving that lukshef-ax1 1449 with nic-mp 1426 can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
renicax  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  -/\  (
( ta  -/\  ( ta  -/\  ta ) ) 
-/\  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )

Proof of Theorem renicax
StepHypRef Expression
1 lukshefth1 1450 . . . 4  |-  ( ( ( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  -/\  ( ta  -/\  ( ta  -/\  ta ) ) )  -/\  ( ph  -/\  ( ch  -/\ 
ps ) ) )
2 lukshefth2 1451 . . . 4  |-  ( ( ( ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  -/\  ( ta  -/\  ( ta  -/\  ta ) ) )  -/\  ( ph  -/\  ( ch  -/\ 
ps ) ) ) 
-/\  ( ( (
ph  -/\  ( ch  -/\  ps ) )  -/\  (
( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  -/\  ( ta  -/\  ( ta  -/\  ta ) ) ) ) 
-/\  ( ( ph  -/\  ( ch  -/\  ps )
)  -/\  ( (
( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  -/\  ( ta  -/\  ( ta  -/\  ta ) ) ) ) ) )
31, 2nic-mp 1426 . . 3  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  -/\  (
( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  -/\  ( ta  -/\  ( ta  -/\  ta ) ) ) )
4 lukshefth2 1451 . . . 4  |-  ( ( ( ta  -/\  ( ta  -/\  ta ) ) 
-/\  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )  -/\  ( ( ( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  -/\  ( ta  -/\  ( ta  -/\  ta ) ) )  -/\  ( ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  -/\  ( ta  -/\  ( ta  -/\  ta ) ) ) ) )
5 lukshef-ax1 1449 . . . 4  |-  ( ( ( ( ta  -/\  ( ta  -/\  ta )
)  -/\  ( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )  -/\  ( ( ( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  -/\  ( ta  -/\  ( ta  -/\  ta ) ) )  -/\  ( ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  -/\  ( ta  -/\  ( ta  -/\  ta ) ) ) ) )  -/\  ( (
( ph  -/\  ( ch 
-/\  ps ) )  -/\  ( ( ph  -/\  ( ch  -/\  ps ) ) 
-/\  ( ph  -/\  ( ch  -/\  ps ) ) ) )  -/\  (
( ( ph  -/\  ( ch  -/\  ps ) ) 
-/\  ( ( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  -/\  ( ta  -/\  ( ta  -/\  ta ) ) ) ) 
-/\  ( ( ( ( ta  -/\  ( ta  -/\  ta ) ) 
-/\  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )  -/\  ( ph  -/\  ( ch  -/\ 
ps ) ) ) 
-/\  ( ( ( ta  -/\  ( ta  -/\ 
ta ) )  -/\  ( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )  -/\  ( ph  -/\  ( ch  -/\ 
ps ) ) ) ) ) ) )
64, 5nic-mp 1426 . . 3  |-  ( ( ( ph  -/\  ( ch  -/\  ps ) ) 
-/\  ( ( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) )  -/\  ( ta  -/\  ( ta  -/\  ta ) ) ) ) 
-/\  ( ( ( ( ta  -/\  ( ta  -/\  ta ) ) 
-/\  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )  -/\  ( ph  -/\  ( ch  -/\ 
ps ) ) ) 
-/\  ( ( ( ta  -/\  ( ta  -/\ 
ta ) )  -/\  ( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )  -/\  ( ph  -/\  ( ch  -/\ 
ps ) ) ) ) )
73, 6nic-mp 1426 . 2  |-  ( ( ( ta  -/\  ( ta  -/\  ta ) ) 
-/\  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )  -/\  ( ph  -/\  ( ch  -/\ 
ps ) ) )
8 lukshefth2 1451 . 2  |-  ( ( ( ( ta  -/\  ( ta  -/\  ta )
)  -/\  ( ( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) )  -/\  ( ph  -/\  ( ch  -/\ 
ps ) ) ) 
-/\  ( ( (
ph  -/\  ( ch  -/\  ps ) )  -/\  (
( ta  -/\  ( ta  -/\  ta ) ) 
-/\  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) ) 
-/\  ( ( ph  -/\  ( ch  -/\  ps )
)  -/\  ( ( ta  -/\  ( ta  -/\  ta ) )  -/\  (
( th  -/\  ch )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) ) ) )
97, 8nic-mp 1426 1  |-  ( (
ph  -/\  ( ch  -/\  ps ) )  -/\  (
( ta  -/\  ( ta  -/\  ta ) ) 
-/\  ( ( th 
-/\  ch )  -/\  (
( ph  -/\  th )  -/\  ( ph  -/\  th )
) ) ) )
Colors of variables: wff set class
Syntax hints:    -/\ wnan 1287
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-an 360  df-nan 1288
  Copyright terms: Public domain W3C validator