adh-minim-ax2

Description: Derivation of ax-2 from adh-minim and ax-mp . Carew Arthur Meredith derived ax-2 inA single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170. However, here we follow the shortened derivation by Ivo Thomas,On Meredith's sole positive axiom, Notre Dame Journal of Formal Logic, volume XV, number 3, July 1974, page 477. Polish prefix notation: CCpCqrCCpqCpr . (Contributed by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		adh-minim-ax2c	$⊢ (φ \to ψ) \to ((φ \to (ψ \to χ)) \to (φ \to χ))$
2		adh-minim-ax1-ax2-lem3	$⊢ ((φ \to ψ) \to ((φ \to (ψ \to χ)) \to (φ \to χ))) \to ((φ \to (ψ \to χ)) \to ((((θ \to τ) \to η) \to ((τ \to (η \to ζ)) \to (τ \to ζ))) \to ((φ \to ψ) \to (φ \to χ))))$
3	1 2	ax-mp	$⊢ (φ \to (ψ \to χ)) \to ((((θ \to τ) \to η) \to ((τ \to (η \to ζ)) \to (τ \to ζ))) \to ((φ \to ψ) \to (φ \to χ)))$
4		adh-minim-ax2-lem6	$⊢ ((φ \to (ψ \to χ)) \to ((((θ \to τ) \to η) \to ((τ \to (η \to ζ)) \to (τ \to ζ))) \to ((φ \to ψ) \to (φ \to χ)))) \to ((φ \to (ψ \to χ)) \to ((φ \to ψ) \to (φ \to χ)))$
5	3 4	ax-mp	$⊢ (φ \to (ψ \to χ)) \to ((φ \to ψ) \to (φ \to χ))$

Metamath Proof Explorer

Theorem adh-minim-ax2

Proof