Theorem nic-ax 1676

Description: Nicod's axiom derived from the standard ones. See Introduction to Mathematical Philosophy by B. Russell, p. 152. Like meredith 1644, the usual axioms can be derived from this and vice versa. Unlike meredith 1644, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g., { nic-ax 1676, nic-mp 1674 } is equivalent to { luk-1 1658, luk-2 1659, luk-3 1660, ax-mp 5 }. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom ($a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nic-ax	⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))

Proof of Theorem nic-ax

Step	Hyp	Ref	Expression
1		nannan 1492	. . . . 5 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ↔ (𝜑 → (𝜒 ∧ 𝜓)))
2	1	biimpi 215	. . . 4 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) → (𝜑 → (𝜒 ∧ 𝜓)))
3		simpl 483	. . . . 5 ⊢ ((𝜒 ∧ 𝜓) → 𝜒)
4	3	imim2i 16	. . . 4 ⊢ ((𝜑 → (𝜒 ∧ 𝜓)) → (𝜑 → 𝜒))
5		imnan 400	. . . . . . 7 ⊢ ((𝜃 → ¬ 𝜒) ↔ ¬ (𝜃 ∧ 𝜒))
6		df-nan 1487	. . . . . . 7 ⊢ ((𝜃 ⊼ 𝜒) ↔ ¬ (𝜃 ∧ 𝜒))
7	5, 6	bitr4i 277	. . . . . 6 ⊢ ((𝜃 → ¬ 𝜒) ↔ (𝜃 ⊼ 𝜒))
8		con3 153	. . . . . . . 8 ⊢ ((𝜑 → 𝜒) → (¬ 𝜒 → ¬ 𝜑))
9	8	imim2d 57	. . . . . . 7 ⊢ ((𝜑 → 𝜒) → ((𝜃 → ¬ 𝜒) → (𝜃 → ¬ 𝜑)))
10		imnan 400	. . . . . . . 8 ⊢ ((𝜑 → ¬ 𝜃) ↔ ¬ (𝜑 ∧ 𝜃))
11		con2b 360	. . . . . . . 8 ⊢ ((𝜃 → ¬ 𝜑) ↔ (𝜑 → ¬ 𝜃))
12		df-nan 1487	. . . . . . . 8 ⊢ ((𝜑 ⊼ 𝜃) ↔ ¬ (𝜑 ∧ 𝜃))
13	10, 11, 12	3bitr4ri 304	. . . . . . 7 ⊢ ((𝜑 ⊼ 𝜃) ↔ (𝜃 → ¬ 𝜑))
14	9, 13	syl6ibr 251	. . . . . 6 ⊢ ((𝜑 → 𝜒) → ((𝜃 → ¬ 𝜒) → (𝜑 ⊼ 𝜃)))
15	7, 14	syl5bir 242	. . . . 5 ⊢ ((𝜑 → 𝜒) → ((𝜃 ⊼ 𝜒) → (𝜑 ⊼ 𝜃)))
16		nanim 1493	. . . . 5 ⊢ (((𝜃 ⊼ 𝜒) → (𝜑 ⊼ 𝜃)) ↔ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))
17	15, 16	sylib 217	. . . 4 ⊢ ((𝜑 → 𝜒) → ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))
18	2, 4, 17	3syl 18	. . 3 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) → ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))
19		pm4.24 564	. . . . 5 ⊢ (𝜏 ↔ (𝜏 ∧ 𝜏))
20	19	biimpi 215	. . . 4 ⊢ (𝜏 → (𝜏 ∧ 𝜏))
21		nannan 1492	. . . 4 ⊢ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ↔ (𝜏 → (𝜏 ∧ 𝜏)))
22	20, 21	mpbir 230	. . 3 ⊢ (𝜏 ⊼ (𝜏 ⊼ 𝜏))
23	18, 22	jctil 520	. 2 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) → ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ∧ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
24		nannan 1492	. 2 ⊢ (((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))) ↔ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) → ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ∧ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))))
25	23, 24	mpbir 230	1 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ⊼ wnan 1486

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 206  df-an 397  df-nan 1487

This theorem is referenced by:  nic-imp  1678  nic-idlem1  1679  nic-idlem2  1680  nic-id  1681  nic-swap  1682  nic-luk1  1694  lukshef-ax1  1697