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Theorem meredith 1400
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 10, where negation and implication are primitive. Here we prove Meredith's axiom from ax-1 7, ax-2 8, and ax-3 9. Then from it we derive the Lukasiewicz axioms luk-1 1415, luk-2 1416, and luk-3 1417. Using these we finally re-derive our axioms as ax1 1426, ax2 1427, and ax3 1428, 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  |-  ( ( ( ( ( ph  ->  ps )  ->  ( -.  ch  ->  -.  th )
)  ->  ch )  ->  ta )  ->  (
( ta  ->  ph )  ->  ( th  ->  ph )
) )

Proof of Theorem meredith
StepHypRef Expression
1 pm2.21 102 . . . . . . 7  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
2 ax-3 9 . . . . . . 7  |-  ( ( -.  ch  ->  -.  th )  ->  ( th  ->  ch ) )
31, 2imim12i 55 . . . . . 6  |-  ( ( ( ph  ->  ps )  ->  ( -.  ch  ->  -.  th ) )  ->  ( -.  ph  ->  ( th  ->  ch ) ) )
43com13 76 . . . . 5  |-  ( th 
->  ( -.  ph  ->  ( ( ( ph  ->  ps )  ->  ( -.  ch  ->  -.  th )
)  ->  ch )
) )
54con1d 118 . . . 4  |-  ( th 
->  ( -.  ( ( ( ph  ->  ps )  ->  ( -.  ch  ->  -.  th ) )  ->  ch )  ->  ph ) )
65com12 29 . . 3  |-  ( -.  ( ( ( ph  ->  ps )  ->  ( -.  ch  ->  -.  th )
)  ->  ch )  ->  ( th  ->  ph )
)
76a1d 24 . 2  |-  ( -.  ( ( ( ph  ->  ps )  ->  ( -.  ch  ->  -.  th )
)  ->  ch )  ->  ( ( ta  ->  ph )  ->  ( th  ->  ph ) ) )
8 ax-1 7 . . 3  |-  ( ta 
->  ( th  ->  ta ) )
98imim1d 71 . 2  |-  ( ta 
->  ( ( ta  ->  ph )  ->  ( th  ->  ph ) ) )
107, 9ja 155 1  |-  ( ( ( ( ( ph  ->  ps )  ->  ( -.  ch  ->  -.  th )
)  ->  ch )  ->  ta )  ->  (
( ta  ->  ph )  ->  ( th  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6
This theorem is referenced by:  merco1  1473
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10
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