adh-minim

Description: A single axiom for minimal implicational calculus, due to Meredith. Other single axioms of the same length are known, but it is thought to be the minimal length. This is the axiom from Carew Arthur Meredith,A single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170. A two-line review by Alonzo Church of this article can be found in The Journal of Symbolic Logic, volume 19, issue 2, June 1954, page 144, https://doi.org/10.2307/2268914 . Known as "HI-1" on Dolph Edward "Ted" Ulrich's web page. In the next 6 lemmas and 3 theorems, ax-1 and ax-2 are derived from this single axiom in 16 detachments (instances of ax-mp ) in total. Polish prefix notation: CCCpqrCsCCqCrtCqt . (Contributed by ADH, 10-Nov-2023)

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

Proof

Step	Hyp	Ref	Expression
1		pm2.04	$⊢ (ψ \to (χ \to τ)) \to (χ \to (ψ \to τ))$
2		adh-jarrsc	$⊢ ((φ \to ψ) \to χ) \to ((χ \to (ψ \to τ)) \to (ψ \to χ))$
3		pm2.04	$⊢ (χ \to (ψ \to τ)) \to (ψ \to (χ \to τ))$
4		ax-2	$⊢ (ψ \to (χ \to τ)) \to ((ψ \to χ) \to (ψ \to τ))$
5		imim2	$⊢ ((ψ \to (χ \to τ)) \to ((ψ \to χ) \to (ψ \to τ))) \to (((χ \to (ψ \to τ)) \to (ψ \to (χ \to τ))) \to ((χ \to (ψ \to τ)) \to ((ψ \to χ) \to (ψ \to τ))))$
6	4 5	ax-mp	$⊢ ((χ \to (ψ \to τ)) \to (ψ \to (χ \to τ))) \to ((χ \to (ψ \to τ)) \to ((ψ \to χ) \to (ψ \to τ)))$
7	3 6	ax-mp	$⊢ (χ \to (ψ \to τ)) \to ((ψ \to χ) \to (ψ \to τ))$
8		ax-2	$⊢ ((χ \to (ψ \to τ)) \to ((ψ \to χ) \to (ψ \to τ))) \to (((χ \to (ψ \to τ)) \to (ψ \to χ)) \to ((χ \to (ψ \to τ)) \to (ψ \to τ)))$
9	7 8	ax-mp	$⊢ ((χ \to (ψ \to τ)) \to (ψ \to χ)) \to ((χ \to (ψ \to τ)) \to (ψ \to τ))$
10		imim2	$⊢ (((χ \to (ψ \to τ)) \to (ψ \to χ)) \to ((χ \to (ψ \to τ)) \to (ψ \to τ))) \to ((((φ \to ψ) \to χ) \to ((χ \to (ψ \to τ)) \to (ψ \to χ))) \to (((φ \to ψ) \to χ) \to ((χ \to (ψ \to τ)) \to (ψ \to τ))))$
11	9 10	ax-mp	$⊢ (((φ \to ψ) \to χ) \to ((χ \to (ψ \to τ)) \to (ψ \to χ))) \to (((φ \to ψ) \to χ) \to ((χ \to (ψ \to τ)) \to (ψ \to τ)))$
12	2 11	ax-mp	$⊢ ((φ \to ψ) \to χ) \to ((χ \to (ψ \to τ)) \to (ψ \to τ))$
13		pm2.04	$⊢ (((φ \to ψ) \to χ) \to ((χ \to (ψ \to τ)) \to (ψ \to τ))) \to ((χ \to (ψ \to τ)) \to (((φ \to ψ) \to χ) \to (ψ \to τ)))$
14	12 13	ax-mp	$⊢ (χ \to (ψ \to τ)) \to (((φ \to ψ) \to χ) \to (ψ \to τ))$
15		imim2	$⊢ ((χ \to (ψ \to τ)) \to (((φ \to ψ) \to χ) \to (ψ \to τ))) \to (((ψ \to (χ \to τ)) \to (χ \to (ψ \to τ))) \to ((ψ \to (χ \to τ)) \to (((φ \to ψ) \to χ) \to (ψ \to τ))))$
16	14 15	ax-mp	$⊢ ((ψ \to (χ \to τ)) \to (χ \to (ψ \to τ))) \to ((ψ \to (χ \to τ)) \to (((φ \to ψ) \to χ) \to (ψ \to τ)))$
17	1 16	ax-mp	$⊢ (ψ \to (χ \to τ)) \to (((φ \to ψ) \to χ) \to (ψ \to τ))$
18		pm2.04	$⊢ ((ψ \to (χ \to τ)) \to (((φ \to ψ) \to χ) \to (ψ \to τ))) \to (((φ \to ψ) \to χ) \to ((ψ \to (χ \to τ)) \to (ψ \to τ)))$
19	17 18	ax-mp	$⊢ ((φ \to ψ) \to χ) \to ((ψ \to (χ \to τ)) \to (ψ \to τ))$
20		ax-1	$⊢ (((φ \to ψ) \to χ) \to ((ψ \to (χ \to τ)) \to (ψ \to τ))) \to (θ \to (((φ \to ψ) \to χ) \to ((ψ \to (χ \to τ)) \to (ψ \to τ))))$
21	19 20	ax-mp	$⊢ θ \to (((φ \to ψ) \to χ) \to ((ψ \to (χ \to τ)) \to (ψ \to τ)))$
22		pm2.04	$⊢ (θ \to (((φ \to ψ) \to χ) \to ((ψ \to (χ \to τ)) \to (ψ \to τ)))) \to (((φ \to ψ) \to χ) \to (θ \to ((ψ \to (χ \to τ)) \to (ψ \to τ))))$
23	21 22	ax-mp	$⊢ ((φ \to ψ) \to χ) \to (θ \to ((ψ \to (χ \to τ)) \to (ψ \to τ)))$

Metamath Proof Explorer

Theorem adh-minim

Proof