Theorem luklem7

Description: Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	luklem7	|- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		luk-1	|- ( ( ph -> ( ps -> ch ) ) -> ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) )
2		luklem5	|- ( ps -> ( ( ps -> ch ) -> ps ) )
3		luk-1	|- ( ( ( ps -> ch ) -> ps ) -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
4	2 3	luklem1	|- ( ps -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
5		luklem6	|- ( ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ps -> ch ) -> ch ) )
6	4 5	luklem1	|- ( ps -> ( ( ps -> ch ) -> ch ) )
7		luk-1	|- ( ( ps -> ( ( ps -> ch ) -> ch ) ) -> ( ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) )
8	6 7	ax-mp	|- ( ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) )
9	1 8	luklem1	|- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )