Metamath Proof Explorer

Theorem adh-minimp-ax2

Description: Derivation of ax-2 from adh-minimp and ax-mp . Polish prefix notation: CCpCqrCCpqCpr . (Contributed by BJ, 4-Apr-2021) (Revised by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	adh-minimp-ax2	$⊢ (φ \to (ψ \to χ)) \to ((φ \to ψ) \to (φ \to χ))$

Proof

Step	Hyp	Ref	Expression
1		adh-minimp-ax2-lem4	$⊢ (φ \to (ψ \to χ)) \to (((φ \to ψ) \to ((φ \to (ψ \to χ)) \to (φ \to χ))) \to ((φ \to ψ) \to (φ \to χ)))$
2		adh-minimp-ax2c	$⊢ (φ \to ψ) \to ((φ \to (ψ \to χ)) \to (φ \to χ))$
3		adh-minimp-ax2-lem4	$⊢ ((φ \to ψ) \to ((φ \to (ψ \to χ)) \to (φ \to χ))) \to (((φ \to (ψ \to χ)) \to (((φ \to ψ) \to ((φ \to (ψ \to χ)) \to (φ \to χ))) \to ((φ \to ψ) \to (φ \to χ)))) \to ((φ \to (ψ \to χ)) \to ((φ \to ψ) \to (φ \to χ))))$
4	2 3	ax-mp	$⊢ ((φ \to (ψ \to χ)) \to (((φ \to ψ) \to ((φ \to (ψ \to χ)) \to (φ \to χ))) \to ((φ \to ψ) \to (φ \to χ)))) \to ((φ \to (ψ \to χ)) \to ((φ \to ψ) \to (φ \to χ)))$
5	1 4	ax-mp	$⊢ (φ \to (ψ \to χ)) \to ((φ \to ψ) \to (φ \to χ))$