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Theorem luklem7 1419
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
luklem7  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )

Proof of Theorem luklem7
StepHypRef Expression
1 luk-1 1410 . 2  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  (
( ( ps  ->  ch )  ->  ch )  ->  ( ph  ->  ch ) ) )
2 luklem5 1417 . . . . 5  |-  ( ps 
->  ( ( ps  ->  ch )  ->  ps )
)
3 luk-1 1410 . . . . 5  |-  ( ( ( ps  ->  ch )  ->  ps )  -> 
( ( ps  ->  ch )  ->  ( ( ps  ->  ch )  ->  ch ) ) )
42, 3luklem1 1413 . . . 4  |-  ( ps 
->  ( ( ps  ->  ch )  ->  ( ( ps  ->  ch )  ->  ch ) ) )
5 luklem6 1418 . . . 4  |-  ( ( ( ps  ->  ch )  ->  ( ( ps 
->  ch )  ->  ch ) )  ->  (
( ps  ->  ch )  ->  ch ) )
64, 5luklem1 1413 . . 3  |-  ( ps 
->  ( ( ps  ->  ch )  ->  ch )
)
7 luk-1 1410 . . 3  |-  ( ( ps  ->  ( ( ps  ->  ch )  ->  ch ) )  ->  (
( ( ( ps 
->  ch )  ->  ch )  ->  ( ph  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) ) )
86, 7ax-mp 8 . 2  |-  ( ( ( ( ps  ->  ch )  ->  ch )  ->  ( ph  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )
91, 8luklem1 1413 1  |-  ( (
ph  ->  ( ps  ->  ch ) )  ->  ( ps  ->  ( ph  ->  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4
This theorem is referenced by:  luklem8  1420  ax2  1422
This theorem was proved from axioms:  ax-mp 8  ax-meredith 1396
  Copyright terms: Public domain W3C validator