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Theorem meredith 1554
 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 1568, luk-2 1569, and luk-3 1570. Using these we finally rederive our axioms as ax1 1579, ax2 1580, and ax3 1581, 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 pm2.21 113 . . . . . . 7 𝜑 → (𝜑𝜓))
2 con4 109 . . . . . . 7 ((¬ 𝜒 → ¬ 𝜃) → (𝜃𝜒))
31, 2imim12i 59 . . . . . 6 (((𝜑𝜓) → (¬ 𝜒 → ¬ 𝜃)) → (¬ 𝜑 → (𝜃𝜒)))
43com13 84 . . . . 5 (𝜃 → (¬ 𝜑 → (((𝜑𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒)))
54con1d 131 . . . 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 79 . 2 (𝜏 → ((𝜏𝜑) → (𝜃𝜑)))
107, 9ja 168 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  1555  merlem2  1556  merlem3  1557  merlem4  1558  merlem5  1559  merlem7  1561  merlem8  1562  merlem9  1563  merlem10  1564  merlem11  1565  merlem13  1567  luk-1  1568  luk-2  1569  merco1  1626
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