Metamath Proof Explorer

Theorem adh-minimp-jarr-lem2

Description: Second lemma for the derivation of jarr , and indirectly ax-1 , a commuted form of ax-2 , and ax-2 proper, from adh-minimp and ax-mp . Polish prefix notation: CCCpqCCCrsCCCtrCsuCruvCqv . (Contributed by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	adh-minimp-jarr-lem2	$⊢ ((φ \to ψ) \to (((χ \to θ) \to (((τ \to χ) \to (θ \to η)) \to (χ \to η))) \to ζ)) \to (ψ \to ζ)$

Proof

Step	Hyp	Ref	Expression
1		adh-minimp	$⊢ ψ \to ((χ \to θ) \to (((τ \to χ) \to (θ \to η)) \to (χ \to η)))$
2		adh-minimp-jarr-imim1-ax2c-lem1	$⊢ (ψ \to ((χ \to θ) \to (((τ \to χ) \to (θ \to η)) \to (χ \to η)))) \to (((φ \to ψ) \to (((χ \to θ) \to (((τ \to χ) \to (θ \to η)) \to (χ \to η))) \to ζ)) \to (ψ \to ζ))$
3	1 2	ax-mp	$⊢ ((φ \to ψ) \to (((χ \to θ) \to (((τ \to χ) \to (θ \to η)) \to (χ \to η))) \to ζ)) \to (ψ \to ζ)$