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

Proof

Step	Hyp	Ref	Expression
1		impsingle-step18	⊢ ( ( ( ( 𝜏 → 𝜂 ) → ( 𝜁 → 𝜂 ) ) → ( ( ( 𝜂 → 𝜎 ) → 𝜏 ) → 𝜌 ) ) → ( 𝜇 → ( ( ( 𝜂 → 𝜎 ) → 𝜏 ) → 𝜌 ) ) )
2		impsingle-step18	⊢ ( ( ( ( 𝜃 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( 𝜑 → 𝜓 ) ) ) )
3		impsingle-step18	⊢ ( ( ( ( ( 𝜃 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( ( ( 𝜏 → 𝜂 ) → ( 𝜁 → 𝜂 ) ) → ( ( ( 𝜂 → 𝜎 ) → 𝜏 ) → 𝜌 ) ) → ( 𝜇 → ( ( ( 𝜂 → 𝜎 ) → 𝜏 ) → 𝜌 ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( 𝜑 → 𝜓 ) ) ) ) )
4	2 3	ax-mp	⊢ ( ( ( ( ( 𝜏 → 𝜂 ) → ( 𝜁 → 𝜂 ) ) → ( ( ( 𝜂 → 𝜎 ) → 𝜏 ) → 𝜌 ) ) → ( 𝜇 → ( ( ( 𝜂 → 𝜎 ) → 𝜏 ) → 𝜌 ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( 𝜑 → 𝜓 ) ) ) )
5	1 4	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜃 ) → ( 𝜑 → 𝜓 ) ) )