Theorem luklem4 1586

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
luklem4	|- ( ( ( ( -. ph -> ph ) -> ph ) -> ps ) -> ps )

Proof of Theorem luklem4

Step	Hyp	Ref	Expression
1		luk-2 1581	. . . 4 |- ( ( -. ( ( -. ph -> ph ) -> ph ) -> ( ( -. ph -> ph ) -> ph ) ) -> ( ( -. ph -> ph ) -> ph ) )
2		luk-2 1581	. . . . 5 |- ( ( -. ph -> ph ) -> ph )
3		luklem3 1585	. . . . 5 |- ( ( ( -. ph -> ph ) -> ph ) -> ( ( ( -. ( ( -. ph -> ph ) -> ph ) -> ( ( -. ph -> ph ) -> ph ) ) -> ( ( -. ph -> ph ) -> ph ) ) -> ( -. ps -> ( ( -. ph -> ph ) -> ph ) ) ) )
4	2, 3	ax-mp 5	. . . 4 |- ( ( ( -. ( ( -. ph -> ph ) -> ph ) -> ( ( -. ph -> ph ) -> ph ) ) -> ( ( -. ph -> ph ) -> ph ) ) -> ( -. ps -> ( ( -. ph -> ph ) -> ph ) ) )
5	1, 4	ax-mp 5	. . 3 |- ( -. ps -> ( ( -. ph -> ph ) -> ph ) )
6		luk-1 1580	. . 3 |- ( ( -. ps -> ( ( -. ph -> ph ) -> ph ) ) -> ( ( ( ( -. ph -> ph ) -> ph ) -> ps ) -> ( -. ps -> ps ) ) )
7	5, 6	ax-mp 5	. 2 |- ( ( ( ( -. ph -> ph ) -> ph ) -> ps ) -> ( -. ps -> ps ) )
8		luk-2 1581	. 2 |- ( ( -. ps -> ps ) -> ps )
9	7, 8	luklem1 1583	1 |- ( ( ( ( -. ph -> ph ) -> ph ) -> ps ) -> 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:  luklem5  1587  luklem6  1588  ax3  1593