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Mirrors > Home > MPE Home > Th. List > merco1 | Structured version Visualization version GIF version |
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 1740 has 21 symbols, sans parentheses. See merco2 1835 for another axiom of equal length. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
merco1 | ⊢ (((((𝜑 → 𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((𝜏 → 𝜑) → (𝜒 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . . . . . 6 ⊢ (¬ 𝜒 → (¬ 𝜃 → ¬ 𝜒)) | |
2 | falim 1674 | . . . . . 6 ⊢ (⊥ → (¬ 𝜃 → ¬ 𝜒)) | |
3 | 1, 2 | ja 175 | . . . . 5 ⊢ ((𝜒 → ⊥) → (¬ 𝜃 → ¬ 𝜒)) |
4 | 3 | imim2i 16 | . . . 4 ⊢ (((𝜑 → 𝜓) → (𝜒 → ⊥)) → ((𝜑 → 𝜓) → (¬ 𝜃 → ¬ 𝜒))) |
5 | 4 | imim1i 63 | . . 3 ⊢ ((((𝜑 → 𝜓) → (¬ 𝜃 → ¬ 𝜒)) → 𝜃) → (((𝜑 → 𝜓) → (𝜒 → ⊥)) → 𝜃)) |
6 | 5 | imim1i 63 | . 2 ⊢ (((((𝜑 → 𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((((𝜑 → 𝜓) → (¬ 𝜃 → ¬ 𝜒)) → 𝜃) → 𝜏)) |
7 | meredith 1740 | . 2 ⊢ (((((𝜑 → 𝜓) → (¬ 𝜃 → ¬ 𝜒)) → 𝜃) → 𝜏) → ((𝜏 → 𝜑) → (𝜒 → 𝜑))) | |
8 | 6, 7 | syl 17 | 1 ⊢ (((((𝜑 → 𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((𝜏 → 𝜑) → (𝜒 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ⊥wfal 1669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 199 df-tru 1660 df-fal 1670 |
This theorem is referenced by: merco1lem1 1813 retbwax2 1815 merco1lem2 1816 merco1lem4 1818 merco1lem5 1819 merco1lem6 1820 merco1lem7 1821 merco1lem10 1825 merco1lem11 1826 merco1lem12 1827 merco1lem13 1828 merco1lem14 1829 merco1lem16 1831 merco1lem17 1832 merco1lem18 1833 retbwax1 1834 |
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