Metamath Proof Explorer

Theorem impsingle-step21

Description: Derivation of impsingle-step21 from ax-mp and impsingle . It is used as a lemma in the proof of imim1 from impsingle . It is Step 21 in Lukasiewicz, where it appears as 'CCCCprqqCCqrCpr' 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-step21	\|- ( ( ( ( ph -> ps ) -> ch ) -> ch ) -> ( ( ch -> ps ) -> ( ph -> ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		impsingle-step15	\|- ( ( ( ch -> ( ph -> ps ) ) -> ( ph -> ps ) ) -> ( ( ch -> ps ) -> ( ph -> ps ) ) )
2		impsingle-step20	\|- ( ( ( ( ch -> ( ph -> ps ) ) -> ( ph -> ps ) ) -> ( ( ch -> ps ) -> ( ph -> ps ) ) ) -> ( ( ( ( ph -> ps ) -> ch ) -> ch ) -> ( ( ch -> ps ) -> ( ph -> ps ) ) ) )
3	1 2	ax-mp	\|- ( ( ( ( ph -> ps ) -> ch ) -> ch ) -> ( ( ch -> ps ) -> ( ph -> ps ) ) )

Step

Hyp

Ref

Expression

impsingle-step15

 |-  ( ( ( ch -> ( ph -> ps ) ) -> ( ph -> ps ) ) -> ( ( ch -> ps ) -> ( ph -> ps ) ) )

impsingle-step20

 |-  ( ( ( ( ch -> ( ph -> ps ) ) -> ( ph -> ps ) ) -> ( ( ch -> ps ) -> ( ph -> ps ) ) ) -> ( ( ( ( ph -> ps ) -> ch ) -> ch ) -> ( ( ch -> ps ) -> ( ph -> ps ) ) ) )

1 2

ax-mp

 |-  ( ( ( ( ph -> ps ) -> ch ) -> ch ) -> ( ( ch -> ps ) -> ( ph -> ps ) ) )