Theorem impsingle-step19

Description: Derivation of impsingle-step19 from ax-mp and impsingle . It is used as a lemma in proofs of imim1 and peirce from impsingle . It is Step 19 in Lukasiewicz, where it appears as 'CCCCspqCrpCCCpqrCsp' 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-step19	|- ( ( ( ( ph -> ps ) -> ch ) -> ( th -> ps ) ) -> ( ( ( ps -> ch ) -> th ) -> ( ph -> ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		impsingle-step18	|- ( ( ( ( ta -> et ) -> ( ze -> et ) ) -> ( ( ( et -> si ) -> ta ) -> rh ) ) -> ( mu -> ( ( ( et -> si ) -> ta ) -> rh ) ) )
2		impsingle-step18	|- ( ( ( ( th -> ps ) -> ( ph -> ps ) ) -> ( ( ( ps -> ch ) -> th ) -> ( ph -> ps ) ) ) -> ( ( ( ( ph -> ps ) -> ch ) -> ( th -> ps ) ) -> ( ( ( ps -> ch ) -> th ) -> ( ph -> ps ) ) ) )
3		impsingle-step18	|- ( ( ( ( ( th -> ps ) -> ( ph -> ps ) ) -> ( ( ( ps -> ch ) -> th ) -> ( ph -> ps ) ) ) -> ( ( ( ( ph -> ps ) -> ch ) -> ( th -> ps ) ) -> ( ( ( ps -> ch ) -> th ) -> ( ph -> ps ) ) ) ) -> ( ( ( ( ( ta -> et ) -> ( ze -> et ) ) -> ( ( ( et -> si ) -> ta ) -> rh ) ) -> ( mu -> ( ( ( et -> si ) -> ta ) -> rh ) ) ) -> ( ( ( ( ph -> ps ) -> ch ) -> ( th -> ps ) ) -> ( ( ( ps -> ch ) -> th ) -> ( ph -> ps ) ) ) ) )
4	2 3	ax-mp	|- ( ( ( ( ( ta -> et ) -> ( ze -> et ) ) -> ( ( ( et -> si ) -> ta ) -> rh ) ) -> ( mu -> ( ( ( et -> si ) -> ta ) -> rh ) ) ) -> ( ( ( ( ph -> ps ) -> ch ) -> ( th -> ps ) ) -> ( ( ( ps -> ch ) -> th ) -> ( ph -> ps ) ) ) )
5	1 4	ax-mp	|- ( ( ( ( ph -> ps ) -> ch ) -> ( th -> ps ) ) -> ( ( ( ps -> ch ) -> th ) -> ( ph -> ps ) ) )