Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  merco1 Structured version   Visualization version   GIF version

Theorem merco1 1714
 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.)
Assertion
Ref Expression
merco1 (((((𝜑𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((𝜏𝜑) → (𝜒𝜑)))

Proof of Theorem merco1
StepHypRef Expression
1 ax-1 6 . . . . . 6 𝜒 → (¬ 𝜃 → ¬ 𝜒))
2 falim 1554 . . . . . 6 (⊥ → (¬ 𝜃 → ¬ 𝜒))
31, 2ja 188 . . . . 5 ((𝜒 → ⊥) → (¬ 𝜃 → ¬ 𝜒))
43imim2i 16 . . . 4 (((𝜑𝜓) → (𝜒 → ⊥)) → ((𝜑𝜓) → (¬ 𝜃 → ¬ 𝜒)))
54imim1i 63 . . 3 ((((𝜑𝜓) → (¬ 𝜃 → ¬ 𝜒)) → 𝜃) → (((𝜑𝜓) → (𝜒 → ⊥)) → 𝜃))
65imim1i 63 . 2 (((((𝜑𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((((𝜑𝜓) → (¬ 𝜃 → ¬ 𝜒)) → 𝜃) → 𝜏))
7 meredith 1642 . 2 (((((𝜑𝜓) → (¬ 𝜃 → ¬ 𝜒)) → 𝜃) → 𝜏) → ((𝜏𝜑) → (𝜒𝜑)))
86, 7syl 17 1 (((((𝜑𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((𝜏𝜑) → (𝜒𝜑)))
 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
 Copyright terms: Public domain W3C validator