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Theorem meredith 1404
 Description: 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 8, where negation and implication are primitive. Here we prove Meredith's axiom from ax-1 5, ax-2 6, and ax-3 7. Then from it we derive the Lukasiewicz axioms luk-1 1420, luk-2 1421, and luk-3 1422. Using these we finally re-derive our axioms as ax1 1431, ax2 1432, and ax3 1433, 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.)
Assertion
Ref Expression
meredith

Proof of Theorem meredith
StepHypRef Expression
1 pm2.21 100 . . . . . . 7
2 ax-3 7 . . . . . . 7
31, 2imim12i 53 . . . . . 6
43com13 74 . . . . 5
54con1d 116 . . . 4
65com12 27 . . 3
76a1d 22 . 2
8 ax-1 5 . . 3
98imim1d 69 . 2
107, 9ja 153 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem is referenced by:  axmeredith  1405  merco1  1478
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