Metamath Proof Explorer

Theorem adh-minim-ax1-ax2-lem2

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

		Ref	Expression
	Assertion	adh-minim-ax1-ax2-lem2	$⊢ (φ \to ((ψ \to ((χ \to (φ \to θ)) \to (χ \to θ))) \to τ)) \to (φ \to τ)$

Proof

Step	Hyp	Ref	Expression
1		adh-minim-ax1-ax2-lem1	$⊢ η \to ((ζ \to ((σ \to ((ρ \to (ζ \to μ)) \to (ρ \to μ))) \to λ)) \to (ζ \to λ))$
2		adh-minim-ax1-ax2-lem1	$⊢ (η \to ((ζ \to ((σ \to ((ρ \to (ζ \to μ)) \to (ρ \to μ))) \to λ)) \to (ζ \to λ))) \to ((φ \to ((ψ \to ((χ \to (φ \to θ)) \to (χ \to θ))) \to τ)) \to (φ \to τ))$
3	1 2	ax-mp	$⊢ (φ \to ((ψ \to ((χ \to (φ \to θ)) \to (χ \to θ))) \to τ)) \to (φ \to τ)$