Metamath Proof Explorer

Theorem adh-minim-ax1-ax2-lem1

Description: First lemma for the derivation of ax-1 and ax-2 from adh-minim and ax-mp . Polish prefix notation: CpCCqCCrCCsCqtCstuCqu . (Contributed by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	adh-minim-ax1-ax2-lem1	$⊢ φ \to ((ψ \to ((χ \to ((θ \to (ψ \to τ)) \to (θ \to τ))) \to η)) \to (ψ \to η))$

Proof

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