MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  luklem6 Unicode version

Theorem luklem6 1418
Description: Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
luklem6  |-  ( (
ph  ->  ( ph  ->  ps ) )  ->  ( ph  ->  ps ) )

Proof of Theorem luklem6
StepHypRef Expression
1 luk-1 1410 . 2  |-  ( (
ph  ->  ( ph  ->  ps ) )  ->  (
( ( ph  ->  ps )  ->  ps )  ->  ( ph  ->  ps ) ) )
2 luklem5 1417 . . . . . 6  |-  ( -.  ( ph  ->  ps )  ->  ( -.  ps  ->  -.  ( ph  ->  ps ) ) )
3 luklem2 1414 . . . . . . 7  |-  ( ( -.  ps  ->  -.  ( ph  ->  ps )
)  ->  ( (
( -.  ps  ->  ps )  ->  ps )  ->  ( ( ph  ->  ps )  ->  ps )
) )
4 luklem4 1416 . . . . . . 7  |-  ( ( ( ( -.  ps  ->  ps )  ->  ps )  ->  ( ( ph  ->  ps )  ->  ps ) )  ->  (
( ph  ->  ps )  ->  ps ) )
53, 4luklem1 1413 . . . . . 6  |-  ( ( -.  ps  ->  -.  ( ph  ->  ps )
)  ->  ( ( ph  ->  ps )  ->  ps ) )
62, 5luklem1 1413 . . . . 5  |-  ( -.  ( ph  ->  ps )  ->  ( ( ph  ->  ps )  ->  ps ) )
7 luk-1 1410 . . . . 5  |-  ( ( -.  ( ph  ->  ps )  ->  ( ( ph  ->  ps )  ->  ps ) )  ->  (
( ( ( ph  ->  ps )  ->  ps )  ->  ( ph  ->  ps ) )  ->  ( -.  ( ph  ->  ps )  ->  ( ph  ->  ps ) ) ) )
86, 7ax-mp 8 . . . 4  |-  ( ( ( ( ph  ->  ps )  ->  ps )  ->  ( ph  ->  ps ) )  ->  ( -.  ( ph  ->  ps )  ->  ( ph  ->  ps ) ) )
9 luk-1 1410 . . . 4  |-  ( ( ( ( ( ph  ->  ps )  ->  ps )  ->  ( ph  ->  ps ) )  ->  ( -.  ( ph  ->  ps )  ->  ( ph  ->  ps ) ) )  -> 
( ( ( -.  ( ph  ->  ps )  ->  ( ph  ->  ps ) )  ->  ( ph  ->  ps ) )  ->  ( ( ( ( ph  ->  ps )  ->  ps )  -> 
( ph  ->  ps )
)  ->  ( ph  ->  ps ) ) ) )
108, 9ax-mp 8 . . 3  |-  ( ( ( -.  ( ph  ->  ps )  ->  ( ph  ->  ps ) )  ->  ( ph  ->  ps ) )  ->  (
( ( ( ph  ->  ps )  ->  ps )  ->  ( ph  ->  ps ) )  ->  ( ph  ->  ps ) ) )
11 luklem4 1416 . . 3  |-  ( ( ( ( -.  ( ph  ->  ps )  -> 
( ph  ->  ps )
)  ->  ( ph  ->  ps ) )  -> 
( ( ( (
ph  ->  ps )  ->  ps )  ->  ( ph  ->  ps ) )  -> 
( ph  ->  ps )
) )  ->  (
( ( ( ph  ->  ps )  ->  ps )  ->  ( ph  ->  ps ) )  ->  ( ph  ->  ps ) ) )
1210, 11ax-mp 8 . 2  |-  ( ( ( ( ph  ->  ps )  ->  ps )  ->  ( ph  ->  ps ) )  ->  ( ph  ->  ps ) )
131, 12luklem1 1413 1  |-  ( (
ph  ->  ( ph  ->  ps ) )  ->  ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4
This theorem is referenced by:  luklem7  1419  ax2  1422
This theorem was proved from axioms:  ax-mp 8  ax-meredith 1396
  Copyright terms: Public domain W3C validator