Description: A single axiom for propositional calculus discovered by C. A. Meredith.
This axiom is worthy of note, due to it having only 19 symbols, not counting parentheses. The more well-known meredith has 21 symbols, sans parentheses.
See merco2 for another axiom of equal length. (Contributed by Anthony Hart, 13-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | merco1 | ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → 𝜃 ) → 𝜏 ) → ( ( 𝜏 → 𝜑 ) → ( 𝜒 → 𝜑 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-1 | ⊢ ( ¬ 𝜒 → ( ¬ 𝜃 → ¬ 𝜒 ) ) | |
| 2 | falim | ⊢ ( ⊥ → ( ¬ 𝜃 → ¬ 𝜒 ) ) | |
| 3 | 1 2 | ja | ⊢ ( ( 𝜒 → ⊥ ) → ( ¬ 𝜃 → ¬ 𝜒 ) ) |
| 4 | 3 | imim2i | ⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → ( ( 𝜑 → 𝜓 ) → ( ¬ 𝜃 → ¬ 𝜒 ) ) ) |
| 5 | 4 | imim1i | ⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( ¬ 𝜃 → ¬ 𝜒 ) ) → 𝜃 ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → 𝜃 ) ) |
| 6 | 5 | imim1i | ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → 𝜃 ) → 𝜏 ) → ( ( ( ( 𝜑 → 𝜓 ) → ( ¬ 𝜃 → ¬ 𝜒 ) ) → 𝜃 ) → 𝜏 ) ) |
| 7 | meredith | ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( ¬ 𝜃 → ¬ 𝜒 ) ) → 𝜃 ) → 𝜏 ) → ( ( 𝜏 → 𝜑 ) → ( 𝜒 → 𝜑 ) ) ) | |
| 8 | 6 7 | syl | ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → 𝜃 ) → 𝜏 ) → ( ( 𝜏 → 𝜑 ) → ( 𝜒 → 𝜑 ) ) ) |