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Theorem meredith 1606
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 1620, luk-2 1621, and luk-3 1622. Using these we finally rederive our axioms as ax1 1631, ax2 1632, and ax3 1633, 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 pm2.21 120 . . . . . . 7 𝜑 → (𝜑𝜓))
2 con4 112 . . . . . . 7 ((¬ 𝜒 → ¬ 𝜃) → (𝜃𝜒))
31, 2imim12i 62 . . . . . 6 (((𝜑𝜓) → (¬ 𝜒 → ¬ 𝜃)) → (¬ 𝜑 → (𝜃𝜒)))
43com13 88 . . . . 5 (𝜃 → (¬ 𝜑 → (((𝜑𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒)))
54con1d 139 . . . 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 173 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  1607  merlem2  1608  merlem3  1609  merlem4  1610  merlem5  1611  merlem7  1613  merlem8  1614  merlem9  1615  merlem10  1616  merlem11  1617  merlem13  1619  luk-1  1620  luk-2  1621  merco1  1678
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