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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 1635 has 21 symbols, sans parentheses. See merco2 1730 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 1550 | . . . . . 6 ⊢ (⊥ → (¬ 𝜃 → ¬ 𝜒)) | |
3 | 1, 2 | ja 186 | . . . . 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 1635 | . 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 1545 |
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 206 df-tru 1536 df-fal 1546 |
This theorem is referenced by: merco1lem1 1708 retbwax2 1710 merco1lem2 1711 merco1lem4 1713 merco1lem5 1714 merco1lem6 1715 merco1lem7 1716 merco1lem10 1720 merco1lem11 1721 merco1lem12 1722 merco1lem13 1723 merco1lem14 1724 merco1lem16 1726 merco1lem17 1727 merco1lem18 1728 retbwax1 1729 |
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