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Mirrors > Home > MPE Home > Th. List > merco1 | Structured version Visualization version GIF version |
Description: A single axiom for
propositional calculus offered by Meredith.
This axiom is worthy of note, due to it having only 19 symbols, not counting parentheses. The more well-known meredith 1566 has 21 symbols, sans parentheses. See merco2 1661 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 1498 | . . . . . 6 ⊢ (⊥ → (¬ 𝜃 → ¬ 𝜒)) | |
3 | 1, 2 | ja 173 | . . . . 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 1566 | . 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 1488 |
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 197 df-tru 1486 df-fal 1489 |
This theorem is referenced by: merco1lem1 1639 retbwax2 1641 merco1lem2 1642 merco1lem4 1644 merco1lem5 1645 merco1lem6 1646 merco1lem7 1647 merco1lem10 1651 merco1lem11 1652 merco1lem12 1653 merco1lem13 1654 merco1lem14 1655 merco1lem16 1657 merco1lem17 1658 merco1lem18 1659 retbwax1 1660 |
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