Theorem impsingle-step22

Description: Derivation of impsingle-step22 from ax-mp and impsingle . It is used as a lemma in proofs of imim1 and peirce from impsingle . It is Step 22 in Lukasiewicz, where it appears as 'Cpp' 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-step22	⊢ ( 𝜑 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		impsingle-step4	⊢ ( ( ( 𝜃 → 𝜇 ) → 𝜃 ) → ( 𝜆 → 𝜃 ) )
2		impsingle-step4	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ( 𝜑 → 𝜑 ) )
3		impsingle-step4	⊢ ( ( ( 𝜑 → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → 𝜑 ) )
4		impsingle	⊢ ( ( ( ( 𝜑 → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → 𝜑 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ( 𝜑 → 𝜑 ) ) → ( ( ( ( 𝜃 → 𝜇 ) → 𝜃 ) → ( 𝜆 → 𝜃 ) ) → ( 𝜑 → 𝜑 ) ) ) )
5	3 4	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ( 𝜑 → 𝜑 ) ) → ( ( ( ( 𝜃 → 𝜇 ) → 𝜃 ) → ( 𝜆 → 𝜃 ) ) → ( 𝜑 → 𝜑 ) ) )
6	2 5	ax-mp	⊢ ( ( ( ( 𝜃 → 𝜇 ) → 𝜃 ) → ( 𝜆 → 𝜃 ) ) → ( 𝜑 → 𝜑 ) )
7	1 6	ax-mp	⊢ ( 𝜑 → 𝜑 )