Theorem merco1 1785

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 1713 has 21 symbols, sans parentheses.

See merco2 1808 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 1645	. . . . . 6 ⊢ (⊥ → (¬ 𝜃 → ¬ 𝜒))
3	1, 2	ja 174	. . . . 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 1713	. 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 1635

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 1633  df-fal 1636

This theorem is referenced by:  merco1lem1  1786  retbwax2  1788  merco1lem2  1789  merco1lem4  1791  merco1lem5  1792  merco1lem6  1793  merco1lem7  1794  merco1lem10  1798  merco1lem11  1799  merco1lem12  1800  merco1lem13  1801  merco1lem14  1802  merco1lem16  1804  merco1lem17  1805  merco1lem18  1806  retbwax1  1807