Theorem impsingle-imim1

Description: Derivation of impsingle-imim1 ( imim1 ) from ax-mp and impsingle . It is step 29 in Lukasiewicz. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	impsingle-imim1	|- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		impsingle-step21	|- ( ( ( ( ph -> ch ) -> ps ) -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) )
2		impsingle-step25	|- ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) )
3		impsingle-step25	|- ( ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) ) -> ( ( ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) ) )
4	2 3	ax-mp	|- ( ( ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) )
5		impsingle-step21	|- ( ( ( ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) ) -> ( ( ( ( ( ph -> ch ) -> ps ) -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) ) )
6	4 5	ax-mp	|- ( ( ( ( ( ph -> ch ) -> ps ) -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) )
7	1 6	ax-mp	|- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) )