adh-minim-ax1-ax2-lem4

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

		Ref	Expression
	Assertion	adh-minim-ax1-ax2-lem4	$⊢ ((φ \to ψ) \to χ) \to ((ψ \to (χ \to θ)) \to (ψ \to θ))$

Proof

Step	Hyp	Ref	Expression
1		adh-minim	$⊢ ((φ \to ψ) \to χ) \to ((η \to ((ζ \to (((φ \to ψ) \to χ) \to σ)) \to (ζ \to σ))) \to ((ψ \to (χ \to θ)) \to (ψ \to θ)))$
2		adh-minim-ax1-ax2-lem2	$⊢ (((φ \to ψ) \to χ) \to ((η \to ((ζ \to (((φ \to ψ) \to χ) \to σ)) \to (ζ \to σ))) \to ((ψ \to (χ \to θ)) \to (ψ \to θ)))) \to (((φ \to ψ) \to χ) \to ((ψ \to (χ \to θ)) \to (ψ \to θ)))$
3	1 2	ax-mp	$⊢ ((φ \to ψ) \to χ) \to ((ψ \to (χ \to θ)) \to (ψ \to θ))$

Metamath Proof Explorer

Theorem adh-minim-ax1-ax2-lem4

Proof