Theorem impsingle-imim1

Description: Derivation of impsingle-imim1 ( imim1 ) from ax-mp and impsingle . It is step 29 in Lukasiewicz. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	impsingle-imim1	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		impsingle-step21	⊢ ( ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )
2		impsingle-step25	⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) )
3		impsingle-step25	⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) )
4	2 3	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) )
5		impsingle-step21	⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) ) )
6	4 5	ax-mp	⊢ ( ( ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) )
7	1 6	ax-mp	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )