Theorem impsingle-step25

Description: Derivation of impsingle-step25 from ax-mp and impsingle . It is used as a lemma in the proof of imim1 from impsingle . It is Step 25 in Lukasiewicz, where it appears as 'CCpqCCCprqq' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	impsingle-step25	|- ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) )

Proof

Step	Hyp	Ref	Expression
1		impsingle-step22	|- ( ( ( ( ph -> ch ) -> ps ) -> ps ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) )
2		impsingle-step20	|- ( ( ( ( ( ph -> ch ) -> ps ) -> ps ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) ) -> ( ( ( ps -> th ) -> ( ph -> ch ) ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) ) )
3	1 2	ax-mp	|- ( ( ( ps -> th ) -> ( ph -> ch ) ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) )
4		impsingle-step8	|- ( ( ( ( ps -> th ) -> ( ph -> ch ) ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) ) -> ( ( ph -> ch ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) ) )
5	3 4	ax-mp	|- ( ( ph -> ch ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) )
6		impsingle-step15	|- ( ( ( ph -> ch ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) ) -> ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) ) )
7	5 6	ax-mp	|- ( ( ph -> ps ) -> ( ( ( ph -> ch ) -> ps ) -> ps ) )