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Theorem meredith 1638
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 5, where negation and implication are primitive. Here we prove Meredith's axiom from ax-1 6, ax-2 7, and ax-3 8. Then from it we derive the Lukasiewicz axioms luk-1 1652, luk-2 1653, and luk-3 1654. Using these we finally rederive our axioms as ax1 1663, ax2 1664, and ax3 1665, 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 (((((𝜑𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒) → 𝜏) → ((𝜏𝜑) → (𝜃𝜑)))

Proof of Theorem meredith
StepHypRef Expression
1 pm2.21 123 . . . . . . 7 𝜑 → (𝜑𝜓))
2 con4 113 . . . . . . 7 ((¬ 𝜒 → ¬ 𝜃) → (𝜃𝜒))
31, 2imim12i 62 . . . . . 6 (((𝜑𝜓) → (¬ 𝜒 → ¬ 𝜃)) → (¬ 𝜑 → (𝜃𝜒)))
43com13 88 . . . . 5 (𝜃 → (¬ 𝜑 → (((𝜑𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒)))
54con1d 147 . . . 4 (𝜃 → (¬ (((𝜑𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒) → 𝜑))
65com12 32 . . 3 (¬ (((𝜑𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒) → (𝜃𝜑))
76a1d 25 . 2 (¬ (((𝜑𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒) → ((𝜏𝜑) → (𝜃𝜑)))
8 ax-1 6 . . 3 (𝜏 → (𝜃𝜏))
98imim1d 82 . 2 (𝜏 → ((𝜏𝜑) → (𝜃𝜑)))
107, 9ja 188 1 (((((𝜑𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒) → 𝜏) → ((𝜏𝜑) → (𝜃𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem is referenced by:  merlem1  1639  merlem2  1640  merlem3  1641  merlem4  1642  merlem5  1643  merlem7  1645  merlem8  1646  merlem9  1647  merlem10  1648  merlem11  1649  merlem13  1651  luk-1  1652  luk-2  1653  merco1  1710
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