Theorem merco1 1710

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 1638 has 21 symbols, sans parentheses.

See merco2 1733 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	⊢ (((((𝜑 → 𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((𝜏 → 𝜑) → (𝜒 → 𝜑)))

Proof of Theorem merco1

Step	Hyp	Ref	Expression
1		ax-1 6	. . . . . 6 ⊢ (¬ 𝜒 → (¬ 𝜃 → ¬ 𝜒))
2		falim 1554	. . . . . 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 1638	. 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)))

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 207  df-tru 1540  df-fal 1550

This theorem is referenced by:  merco1lem1  1711  retbwax2  1713  merco1lem2  1714  merco1lem4  1716  merco1lem5  1717  merco1lem6  1718  merco1lem7  1719  merco1lem10  1723  merco1lem11  1724  merco1lem12  1725  merco1lem13  1726  merco1lem14  1727  merco1lem16  1729  merco1lem17  1730  merco1lem18  1731  retbwax1  1732