Theorem nic-ax 1668

Description: Nicod's axiom derived from the standard ones. See Introduction to Mathematical Philosophy by B. Russell, p. 152. Like meredith 1636, the usual axioms can be derived from this and vice versa. Unlike meredith 1636, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g., { nic-ax 1668, nic-mp 1666 } is equivalent to { luk-1 1650, luk-2 1651, luk-3 1652, 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 1491	. . . . 5 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ↔ (𝜑 → (𝜒 ∧ 𝜓)))
2	1	biimpi 215	. . . 4 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) → (𝜑 → (𝜒 ∧ 𝜓)))
3		simpl 482	. . . . 5 ⊢ ((𝜒 ∧ 𝜓) → 𝜒)
4	3	imim2i 16	. . . 4 ⊢ ((𝜑 → (𝜒 ∧ 𝜓)) → (𝜑 → 𝜒))
5		imnan 399	. . . . . . 7 ⊢ ((𝜃 → ¬ 𝜒) ↔ ¬ (𝜃 ∧ 𝜒))
6		df-nan 1486	. . . . . . 7 ⊢ ((𝜃 ⊼ 𝜒) ↔ ¬ (𝜃 ∧ 𝜒))
7	5, 6	bitr4i 278	. . . . . 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 399	. . . . . . . 8 ⊢ ((𝜑 → ¬ 𝜃) ↔ ¬ (𝜑 ∧ 𝜃))
11		con2b 359	. . . . . . . 8 ⊢ ((𝜃 → ¬ 𝜑) ↔ (𝜑 → ¬ 𝜃))
12		df-nan 1486	. . . . . . . 8 ⊢ ((𝜑 ⊼ 𝜃) ↔ ¬ (𝜑 ∧ 𝜃))
13	10, 11, 12	3bitr4ri 304	. . . . . . 7 ⊢ ((𝜑 ⊼ 𝜃) ↔ (𝜃 → ¬ 𝜑))
14	9, 13	imbitrrdi 251	. . . . . 6 ⊢ ((𝜑 → 𝜒) → ((𝜃 → ¬ 𝜒) → (𝜑 ⊼ 𝜃)))
15	7, 14	biimtrrid 242	. . . . 5 ⊢ ((𝜑 → 𝜒) → ((𝜃 ⊼ 𝜒) → (𝜑 ⊼ 𝜃)))
16		nanim 1492	. . . . 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		pm4.24 563	. . . . 5 ⊢ (𝜏 ↔ (𝜏 ∧ 𝜏))
20	19	biimpi 215	. . . 4 ⊢ (𝜏 → (𝜏 ∧ 𝜏))
21		nannan 1491	. . . 4 ⊢ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ↔ (𝜏 → (𝜏 ∧ 𝜏)))
22	20, 21	mpbir 230	. . 3 ⊢ (𝜏 ⊼ (𝜏 ⊼ 𝜏))
23	18, 22	jctil 519	. 2 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) → ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ∧ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
24		nannan 1491	. 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 395   ⊼ wnan 1485

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 396  df-nan 1486

This theorem is referenced by:  nic-imp  1670  nic-idlem1  1671  nic-idlem2  1672  nic-id  1673  nic-swap  1674  nic-luk1  1686  lukshef-ax1  1689