Theorem impsingle-step15

Description: Derivation of impsingle-step15 from ax-mp and impsingle . It is used as a lemma in proofs of imim1 and peirce from impsingle . It is Step 15 in Lukasiewicz, where it appears as 'CCCrqCspCCrpCsp' 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-step15	⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		impsingle	⊢ ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) )
2		impsingle	⊢ ( ( ( 𝜏 → 𝜎 ) → 𝜌 ) → ( ( 𝜌 → 𝜏 ) → ( 𝜇 → 𝜏 ) ) )
3		impsingle	⊢ ( ( ( ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) → 𝜂 ) → ( ( 𝜃 → 𝜆 ) → 𝜑 ) ) → ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) )
4		impsingle	⊢ ( ( ( ( 𝜒 → 𝜃 ) → 𝜁 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) )
5		impsingle-step8	⊢ ( ( ( ( ( 𝜒 → 𝜃 ) → 𝜁 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) )
6	4 5	ax-mp	⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) )
7		impsingle	⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → 𝜑 ) → ( ( 𝜃 → 𝜆 ) → 𝜑 ) ) )
8	6 7	ax-mp	⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → 𝜑 ) → ( ( 𝜃 → 𝜆 ) → 𝜑 ) )
9		impsingle	⊢ ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → 𝜑 ) → ( ( 𝜃 → 𝜆 ) → 𝜑 ) ) → ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) → ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) ) )
10	8 9	ax-mp	⊢ ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) → ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) )
11		impsingle	⊢ ( ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) → ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) ) → ( ( ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) → ( ( 𝜃 → 𝜆 ) → 𝜑 ) ) → ( ( ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) → 𝜂 ) → ( ( 𝜃 → 𝜆 ) → 𝜑 ) ) ) )
12	10 11	ax-mp	⊢ ( ( ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) → ( ( 𝜃 → 𝜆 ) → 𝜑 ) ) → ( ( ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) → 𝜂 ) → ( ( 𝜃 → 𝜆 ) → 𝜑 ) ) )
13		impsingle	⊢ ( ( ( ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) → ( ( 𝜃 → 𝜆 ) → 𝜑 ) ) → ( ( ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) → 𝜂 ) → ( ( 𝜃 → 𝜆 ) → 𝜑 ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) → 𝜂 ) → ( ( 𝜃 → 𝜆 ) → 𝜑 ) ) → ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) ) → ( ( ( ( 𝜏 → 𝜎 ) → 𝜌 ) → ( ( 𝜌 → 𝜏 ) → ( 𝜇 → 𝜏 ) ) ) → ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) ) ) )
14	12 13	ax-mp	⊢ ( ( ( ( ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) → 𝜂 ) → ( ( 𝜃 → 𝜆 ) → 𝜑 ) ) → ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) ) → ( ( ( ( 𝜏 → 𝜎 ) → 𝜌 ) → ( ( 𝜌 → 𝜏 ) → ( 𝜇 → 𝜏 ) ) ) → ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) ) )
15	3 14	ax-mp	⊢ ( ( ( ( 𝜏 → 𝜎 ) → 𝜌 ) → ( ( 𝜌 → 𝜏 ) → ( 𝜇 → 𝜏 ) ) ) → ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) ) )
16	2 15	ax-mp	⊢ ( ( ( ( 𝜃 → 𝜆 ) → 𝜑 ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) ) )
17	1 16	ax-mp	⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 → 𝜃 ) → ( 𝜒 → 𝜃 ) ) )