Carew Meredith's sole axiom for propositional calculus. This amazing formula is thought to be the shortest possible single axiom for propositional calculus with inference rule [ax-mp], where negation and implication are primitive. Here we prove Meredith's axiom from [ax-1], [ax-2], and [ax-3]. Then from it we derive the Lukasiewicz axioms luk-1, luk-2, and luk-3. Using these we finally rederive our axioms as ax1, ax2, and ax3, thus proving the equivalence of all three systems. C. A. Meredith, "Single Axioms for the Systems (C,N), (C,O) and (A,N) of the Two-Valued Propositional Calculus", The Journal of Computing Systems vol. 1 (1953), pp. 155-164. Meredith claimed to be close to a proof that this axiom is the shortest possible, but the proof was apparently never completed.

An obscure Irish lecturer, Meredith (1904-1976) became enamored with logic somewhat late in life after attending talks by Lukasiewicz and produced many remarkable results such as this axiom. From his obituary: "He did logic whenever time and opportunity presented themselves, and he did it on whatever materials came to hand: in a pub, his favored pint of porter within reach, he would use the inside of cigarette packs to write proofs for logical colleagues." (Contributed by NM, 14-Dec-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Wolf Lammen, 28-May-2013.)