Metamath Proof Explorer

Theorem adh-minim-ax2-lem5

Description: Fifth lemma for the derivation of ax-2 from adh-minim and ax-mp . Polish prefix notation: CpCCCqrsCCrCstCrt . (Contributed by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	adh-minim-ax2-lem5	$⊢ φ \to (((ψ \to χ) \to θ) \to ((χ \to (θ \to τ)) \to (χ \to τ)))$

Proof

Step	Hyp	Ref	Expression
1		adh-minim-ax1-ax2-lem4	$⊢ ((ψ \to χ) \to θ) \to ((χ \to (θ \to τ)) \to (χ \to τ))$
2		adh-minim-ax1	$⊢ (((ψ \to χ) \to θ) \to ((χ \to (θ \to τ)) \to (χ \to τ))) \to (φ \to (((ψ \to χ) \to θ) \to ((χ \to (θ \to τ)) \to (χ \to τ))))$
3	1 2	ax-mp	$⊢ φ \to (((ψ \to χ) \to θ) \to ((χ \to (θ \to τ)) \to (χ \to τ)))$