Theorem impsingle-step8

Description: Derivation of impsingle-step8 from ax-mp and impsingle . It is used as a lemma in proofs of ax-1 imim1 and peirce from impsingle . It is Step 8 in Lukasiewicz, where it appears as 'CCCsqpCqp' 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-step8	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		impsingle	⊢ ( ( ( 𝜏 → 𝜂 ) → 𝜁 ) → ( ( 𝜁 → 𝜏 ) → ( 𝜎 → 𝜏 ) ) )
2		impsingle	⊢ ( ( ( 𝜒 → 𝜃 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) )
3		impsingle	⊢ ( ( ( 𝜓 → 𝜃 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) )
4		impsingle	⊢ ( ( ( 𝜓 → 𝜒 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) )
5		impsingle	⊢ ( ( ( ( 𝜓 → 𝜒 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜃 ) → ( 𝜓 → 𝜒 ) ) ) )
6	4 5	ax-mp	⊢ ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜃 ) → ( 𝜓 → 𝜒 ) ) )
7		impsingle	⊢ ( ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜃 ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ( ( 𝜓 → 𝜃 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ( ( 𝜏 → 𝜂 ) → 𝜁 ) → ( ( 𝜁 → 𝜏 ) → ( 𝜎 → 𝜏 ) ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) )
8	6 7	ax-mp	⊢ ( ( ( ( 𝜓 → 𝜃 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ( ( 𝜏 → 𝜂 ) → 𝜁 ) → ( ( 𝜁 → 𝜏 ) → ( 𝜎 → 𝜏 ) ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) )
9	3 8	ax-mp	⊢ ( ( ( ( 𝜏 → 𝜂 ) → 𝜁 ) → ( ( 𝜁 → 𝜏 ) → ( 𝜎 → 𝜏 ) ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) )
10	1 9	ax-mp	⊢ ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) )
11		impsingle	⊢ ( ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) )
12	10 11	ax-mp	⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) )
13		impsingle	⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜒 → 𝜃 ) → ( 𝜑 → 𝜓 ) ) ) )
14	12 13	ax-mp	⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜒 → 𝜃 ) → ( 𝜑 → 𝜓 ) ) )
15		impsingle	⊢ ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜒 → 𝜃 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ( ( 𝜒 → 𝜃 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ( ( 𝜏 → 𝜂 ) → 𝜁 ) → ( ( 𝜁 → 𝜏 ) → ( 𝜎 → 𝜏 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) ) )
16	14 15	ax-mp	⊢ ( ( ( ( 𝜒 → 𝜃 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ( ( 𝜏 → 𝜂 ) → 𝜁 ) → ( ( 𝜁 → 𝜏 ) → ( 𝜎 → 𝜏 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) )
17	2 16	ax-mp	⊢ ( ( ( ( 𝜏 → 𝜂 ) → 𝜁 ) → ( ( 𝜁 → 𝜏 ) → ( 𝜎 → 𝜏 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) )
18	1 17	ax-mp	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) )