Metamath Proof Explorer

Theorem adh-minim-ax2-lem6

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

		Ref	Expression
	Assertion	adh-minim-ax2-lem6	$⊢ (φ \to ((((ψ \to χ) \to θ) \to ((χ \to (θ \to τ)) \to (χ \to τ))) \to η)) \to (φ \to η)$

Proof

Step	Hyp	Ref	Expression
1		adh-minim-ax2-lem5	$⊢ (ζ \to φ) \to (((ψ \to χ) \to θ) \to ((χ \to (θ \to τ)) \to (χ \to τ)))$
2		adh-minim-ax1-ax2-lem4	$⊢ ((ζ \to φ) \to (((ψ \to χ) \to θ) \to ((χ \to (θ \to τ)) \to (χ \to τ)))) \to ((φ \to ((((ψ \to χ) \to θ) \to ((χ \to (θ \to τ)) \to (χ \to τ))) \to η)) \to (φ \to η))$
3	1 2	ax-mp	$⊢ (φ \to ((((ψ \to χ) \to θ) \to ((χ \to (θ \to τ)) \to (χ \to τ))) \to η)) \to (φ \to η)$