Metamath Proof Explorer

Theorem impsingle-step21

Description: Derivation of impsingle-step21 from ax-mp and impsingle . It is used as a lemma in the proof of imim1 from impsingle . It is Step 21 in Lukasiewicz, where it appears as 'CCCCprqqCCqrCpr' 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-step21	$⊢ (((φ \to ψ) \to χ) \to χ) \to ((χ \to ψ) \to (φ \to ψ))$

Proof

Step	Hyp	Ref	Expression
1		impsingle-step15	$⊢ ((χ \to (φ \to ψ)) \to (φ \to ψ)) \to ((χ \to ψ) \to (φ \to ψ))$
2		impsingle-step20	$⊢ (((χ \to (φ \to ψ)) \to (φ \to ψ)) \to ((χ \to ψ) \to (φ \to ψ))) \to ((((φ \to ψ) \to χ) \to χ) \to ((χ \to ψ) \to (φ \to ψ)))$
3	1 2	ax-mp	$⊢ (((φ \to ψ) \to χ) \to χ) \to ((χ \to ψ) \to (φ \to ψ))$