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	⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		impsingle-step22	⊢ ( ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) )
2		impsingle-step20	⊢ ( ( ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) → ( ( ( 𝜓 → 𝜃 ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) )
3	1 2	ax-mp	⊢ ( ( ( 𝜓 → 𝜃 ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) )
4		impsingle-step8	⊢ ( ( ( ( 𝜓 → 𝜃 ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) )
5	3 4	ax-mp	⊢ ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) )
6		impsingle-step15	⊢ ( ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) )
7	5 6	ax-mp	⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) )