Theorem ax2

Description: Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	ax2	|- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		luklem7	|- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )
2		luklem8	|- ( ( ps -> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) )
3		luklem6	|- ( ( ph -> ( ph -> ch ) ) -> ( ph -> ch ) )
4		luklem8	|- ( ( ( ph -> ( ph -> ch ) ) -> ( ph -> ch ) ) -> ( ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
5	3 4	ax-mp	|- ( ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
6	2 5	luklem1	|- ( ( ps -> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
7	1 6	luklem1	|- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )