Theorem luklem6 1588

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

Step	Hyp	Ref	Expression
1		luk-1 1580	. 2 |- ( ( ph -> ( ph -> ps ) ) -> ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) )
2		luklem5 1587	. . . . . 6 |- ( -. ( ph -> ps ) -> ( -. ps -> -. ( ph -> ps ) ) )
3		luklem2 1584	. . . . . . 7 |- ( ( -. ps -> -. ( ph -> ps ) ) -> ( ( ( -. ps -> ps ) -> ps ) -> ( ( ph -> ps ) -> ps ) ) )
4		luklem4 1586	. . . . . . 7 |- ( ( ( ( -. ps -> ps ) -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ps ) )
5	3, 4	luklem1 1583	. . . . . 6 |- ( ( -. ps -> -. ( ph -> ps ) ) -> ( ( ph -> ps ) -> ps ) )
6	2, 5	luklem1 1583	. . . . 5 |- ( -. ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) )
7		luk-1 1580	. . . . 5 |- ( ( -. ( ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) -> ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( -. ( ph -> ps ) -> ( ph -> ps ) ) ) )
8	6, 7	ax-mp 5	. . . 4 |- ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( -. ( ph -> ps ) -> ( ph -> ps ) ) )
9		luk-1 1580	. . . 4 |- ( ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( -. ( ph -> ps ) -> ( ph -> ps ) ) ) -> ( ( ( -. ( ph -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) ) -> ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) ) ) )
10	8, 9	ax-mp 5	. . 3 |- ( ( ( -. ( ph -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) ) -> ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) ) )
11		luklem4 1586	. . 3 |- ( ( ( ( -. ( ph -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) ) -> ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) ) ) -> ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) ) )
12	10, 11	ax-mp 5	. 2 |- ( ( ( ( ph -> ps ) -> ps ) -> ( ph -> ps ) ) -> ( ph -> ps ) )
13	1, 12	luklem1 1583	1 |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) )

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:  luklem7  1589  ax2  1592