Theorem nic-ax 1670

Description: Nicod's axiom derived from the standard ones. See Introduction to Mathematical Philosophy by B. Russell, p. 152. Like meredith 1638, the usual axioms can be derived from this and vice versa. Unlike meredith 1638, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g. { nic-ax 1670, nic-mp 1668 } is equivalent to { luk-1 1652, luk-2 1653, luk-3 1654, 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 1486	. . . . 5 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ↔ (𝜑 → (𝜒 ∧ 𝜓)))
2	1	biimpi 218	. . . 4 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) → (𝜑 → (𝜒 ∧ 𝜓)))
3		simpl 485	. . . . 5 ⊢ ((𝜒 ∧ 𝜓) → 𝜒)
4	3	imim2i 16	. . . 4 ⊢ ((𝜑 → (𝜒 ∧ 𝜓)) → (𝜑 → 𝜒))
5		imnan 402	. . . . . . 7 ⊢ ((𝜃 → ¬ 𝜒) ↔ ¬ (𝜃 ∧ 𝜒))
6		df-nan 1481	. . . . . . 7 ⊢ ((𝜃 ⊼ 𝜒) ↔ ¬ (𝜃 ∧ 𝜒))
7	5, 6	bitr4i 280	. . . . . 6 ⊢ ((𝜃 → ¬ 𝜒) ↔ (𝜃 ⊼ 𝜒))
8		con3 156	. . . . . . . 8 ⊢ ((𝜑 → 𝜒) → (¬ 𝜒 → ¬ 𝜑))
9	8	imim2d 57	. . . . . . 7 ⊢ ((𝜑 → 𝜒) → ((𝜃 → ¬ 𝜒) → (𝜃 → ¬ 𝜑)))
10		imnan 402	. . . . . . . 8 ⊢ ((𝜑 → ¬ 𝜃) ↔ ¬ (𝜑 ∧ 𝜃))
11		con2b 362	. . . . . . . 8 ⊢ ((𝜃 → ¬ 𝜑) ↔ (𝜑 → ¬ 𝜃))
12		df-nan 1481	. . . . . . . 8 ⊢ ((𝜑 ⊼ 𝜃) ↔ ¬ (𝜑 ∧ 𝜃))
13	10, 11, 12	3bitr4ri 306	. . . . . . 7 ⊢ ((𝜑 ⊼ 𝜃) ↔ (𝜃 → ¬ 𝜑))
14	9, 13	syl6ibr 254	. . . . . 6 ⊢ ((𝜑 → 𝜒) → ((𝜃 → ¬ 𝜒) → (𝜑 ⊼ 𝜃)))
15	7, 14	syl5bir 245	. . . . 5 ⊢ ((𝜑 → 𝜒) → ((𝜃 ⊼ 𝜒) → (𝜑 ⊼ 𝜃)))
16		nanim 1487	. . . . 5 ⊢ (((𝜃 ⊼ 𝜒) → (𝜑 ⊼ 𝜃)) ↔ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))
17	15, 16	sylib 220	. . . 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		pm4.24 566	. . . . 5 ⊢ (𝜏 ↔ (𝜏 ∧ 𝜏))
20	19	biimpi 218	. . . 4 ⊢ (𝜏 → (𝜏 ∧ 𝜏))
21		nannan 1486	. . . 4 ⊢ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ↔ (𝜏 → (𝜏 ∧ 𝜏)))
22	20, 21	mpbir 233	. . 3 ⊢ (𝜏 ⊼ (𝜏 ⊼ 𝜏))
23	18, 22	jctil 522	. 2 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) → ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ∧ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
24		nannan 1486	. 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 233	1 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ⊼ wnan 1480

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-an 399  df-nan 1481

This theorem is referenced by:  nic-imp  1672  nic-idlem1  1673  nic-idlem2  1674  nic-id  1675  nic-swap  1676  nic-luk1  1688  lukshef-ax1  1691