Theorem luklem2 1584

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

Proof of Theorem luklem2

Step	Hyp	Ref	Expression
1		luk-1 1580	. . 3 |- ( ( ph -> -. ps ) -> ( ( -. ps -> ch ) -> ( ph -> ch ) ) )
2		luk-3 1582	. . . 4 |- ( ps -> ( -. ps -> ch ) )
3		luk-1 1580	. . . 4 |- ( ( ps -> ( -. ps -> ch ) ) -> ( ( ( -. ps -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) )
4	2, 3	ax-mp 5	. . 3 |- ( ( ( -. ps -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) )
5	1, 4	luklem1 1583	. 2 |- ( ( ph -> -. ps ) -> ( ps -> ( ph -> ch ) ) )
6		luk-1 1580	. 2 |- ( ( ps -> ( ph -> ch ) ) -> ( ( ( ph -> ch ) -> th ) -> ( ps -> th ) ) )
7	5, 6	luklem1 1583	1 |- ( ( ph -> -. ps ) -> ( ( ( ph -> ch ) -> th ) -> ( ps -> th ) ) )

Syntax hints:   -. wn 3   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  luklem3  1585  luklem6  1588  ax3  1593