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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 1642 has 21 symbols, sans parentheses. See merco2 1737 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 1554 | . . . . . 6 ⊢ (⊥ → (¬ 𝜃 → ¬ 𝜒)) | |
3 | 1, 2 | ja 188 | . . . . 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 1642 | . 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 1549 |
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 209 df-tru 1540 df-fal 1550 |
This theorem is referenced by: merco1lem1 1715 retbwax2 1717 merco1lem2 1718 merco1lem4 1720 merco1lem5 1721 merco1lem6 1722 merco1lem7 1723 merco1lem10 1727 merco1lem11 1728 merco1lem12 1729 merco1lem13 1730 merco1lem14 1731 merco1lem16 1733 merco1lem17 1734 merco1lem18 1735 retbwax1 1736 |
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