Theorem impsingle-step22

Description: Derivation of impsingle-step22 from ax-mp and impsingle . It is used as a lemma in proofs of imim1 and peirce from impsingle . It is Step 22 in Lukasiewicz, where it appears as 'Cpp' 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-step22	|- ( ph -> ph )

Proof

Step	Hyp	Ref	Expression
1		impsingle-step4	|- ( ( ( th -> mu ) -> th ) -> ( la -> th ) )
2		impsingle-step4	|- ( ( ( ph -> ps ) -> ph ) -> ( ph -> ph ) )
3		impsingle-step4	|- ( ( ( ph -> ph ) -> ph ) -> ( ( ph -> ps ) -> ph ) )
4		impsingle	|- ( ( ( ( ph -> ph ) -> ph ) -> ( ( ph -> ps ) -> ph ) ) -> ( ( ( ( ph -> ps ) -> ph ) -> ( ph -> ph ) ) -> ( ( ( ( th -> mu ) -> th ) -> ( la -> th ) ) -> ( ph -> ph ) ) ) )
5	3 4	ax-mp	|- ( ( ( ( ph -> ps ) -> ph ) -> ( ph -> ph ) ) -> ( ( ( ( th -> mu ) -> th ) -> ( la -> th ) ) -> ( ph -> ph ) ) )
6	2 5	ax-mp	|- ( ( ( ( th -> mu ) -> th ) -> ( la -> th ) ) -> ( ph -> ph ) )
7	1 6	ax-mp	|- ( ph -> ph )