Theorem impsingle-step18

Description: Derivation of impsingle-step18 from ax-mp and impsingle . It is used as a lemma in proofs of imim1 and peirce from impsingle . It is Step 18 in Lukasiewicz, where it appears as 'CCCCrpCspCCCpqrtCuCCCpqrt' 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-step18	⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜓 ) ) → ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) ) → ( 𝜂 → ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		impsingle	⊢ ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜓 ) ) )
2		impsingle	⊢ ( ( ( ( 𝜒 → 𝜓 ) → 𝜌 ) → ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) ) → ( ( ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) → ( 𝜒 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜓 ) ) ) )
3		impsingle-step8	⊢ ( ( ( ( ( 𝜒 → 𝜓 ) → 𝜌 ) → ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) ) → ( ( ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) → ( 𝜒 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜓 ) ) ) ) → ( ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) → ( ( ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) → ( 𝜒 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜓 ) ) ) ) )
4	2 3	ax-mp	⊢ ( ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) → ( ( ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) → ( 𝜒 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜓 ) ) ) )
5		impsingle-step15	⊢ ( ( ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) → ( ( ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) → ( 𝜒 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜓 ) ) ) ) → ( ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜓 ) ) ) → ( ( ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) → ( 𝜒 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜓 ) ) ) ) )
6	4 5	ax-mp	⊢ ( ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜓 ) ) ) → ( ( ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) → ( 𝜒 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜓 ) ) ) )
7	1 6	ax-mp	⊢ ( ( ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) → ( 𝜒 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜓 ) ) )
8		impsingle	⊢ ( ( ( ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) → ( 𝜒 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜓 ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜓 ) ) → ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) ) → ( 𝜂 → ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) ) ) )
9	7 8	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜓 ) ) → ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) ) → ( 𝜂 → ( ( ( 𝜓 → 𝜃 ) → 𝜑 ) → 𝜏 ) ) )