Theorem luklem2 1066

Description: Used to rederive standard propositional axioms from Lukasiewicz'.

Assertion

Ref	Expression
luklem2	|- ((ph -> -. ps) -> (((ph -> ch) -> th) -> (ps -> th)))

Proof of Theorem luklem2

Step	Hyp	Ref	Expression
1		luk-1 1062	. . 3 |- ((ph -> -. ps) -> ((-. ps -> ch) -> (ph -> ch)))
2		luk-3 1064	. . . 4 |- (ps -> (-. ps -> ch))
3		luk-1 1062	. . . 4 |- ((ps -> (-. ps -> ch)) -> (((-. ps -> ch) -> (ph -> ch)) -> (ps -> (ph -> ch))))
4	2, 3	ax-mp 7	. . 3 |- (((-. ps -> ch) -> (ph -> ch)) -> (ps -> (ph -> ch)))
5	1, 4	luklem1 1065	. 2 |- ((ph -> -. ps) -> (ps -> (ph -> ch)))
6		luk-1 1062	. 2 |- ((ps -> (ph -> ch)) -> (((ph -> ch) -> th) -> (ps -> th)))
7	5, 6	luklem1 1065	1 |- ((ph -> -. ps) -> (((ph -> ch) -> th) -> (ps -> th)))

Syntax hints:  -. wn 2   -> wi 3

This theorem is referenced by:  luklem3 1067  luklem6 1070  ax3 1075

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 163  df-an 241

Copyright terms: Public domain