Theorem impsingle-step4

Description: Derivation of impsingle-step4 from ax-mp and impsingle . It is used as a lemma in proofs of imim1 and peirce from impsingle . It is Step 4 in Lukasiewicz, where it appears as 'CCCpqpCsp' 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-step4	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ( 𝜒 → 𝜑 ) )

Proof

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