Theorem meredith 1643

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 1657, luk-2 1658, and luk-3 1659. Using these we finally rederive our axioms as ax1 1668, ax2 1669, and ax3 1670, 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	⊢ (((((𝜑 → 𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒) → 𝜏) → ((𝜏 → 𝜑) → (𝜃 → 𝜑)))

Proof of Theorem meredith

Step	Hyp	Ref	Expression
1		pm2.21 123	. . . . . . 7 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
2		con4 113	. . . . . . 7 ⊢ ((¬ 𝜒 → ¬ 𝜃) → (𝜃 → 𝜒))
3	1, 2	imim12i 62	. . . . . 6 ⊢ (((𝜑 → 𝜓) → (¬ 𝜒 → ¬ 𝜃)) → (¬ 𝜑 → (𝜃 → 𝜒)))
4	3	com13 88	. . . . 5 ⊢ (𝜃 → (¬ 𝜑 → (((𝜑 → 𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒)))
5	4	con1d 147	. . . 4 ⊢ (𝜃 → (¬ (((𝜑 → 𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒) → 𝜑))
6	5	com12 32	. . 3 ⊢ (¬ (((𝜑 → 𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒) → (𝜃 → 𝜑))
7	6	a1d 25	. 2 ⊢ (¬ (((𝜑 → 𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒) → ((𝜏 → 𝜑) → (𝜃 → 𝜑)))
8		ax-1 6	. . 3 ⊢ (𝜏 → (𝜃 → 𝜏))
9	8	imim1d 82	. 2 ⊢ (𝜏 → ((𝜏 → 𝜑) → (𝜃 → 𝜑)))
10	7, 9	ja 189	1 ⊢ (((((𝜑 → 𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒) → 𝜏) → ((𝜏 → 𝜑) → (𝜃 → 𝜑)))

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  1644  merlem2  1645  merlem3  1646  merlem4  1647  merlem5  1648  merlem7  1650  merlem8  1651  merlem9  1652  merlem10  1653  merlem11  1654  merlem13  1656  luk-1  1657  luk-2  1658  merco1  1715