Theorem nic-dfim 1669

Description: This theorem "defines" implication in terms of 'nand'. Analogous to nanim 1498. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition ($a statement). (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nic-dfim	⊢ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 → 𝜓)) ⊼ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 ⊼ (𝜓 ⊼ 𝜓))) ⊼ ((𝜑 → 𝜓) ⊼ (𝜑 → 𝜓))))

Proof of Theorem nic-dfim

Step	Hyp	Ref	Expression
1		nanim 1498	. . 3 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ⊼ (𝜓 ⊼ 𝜓)))
2	1	bicomi 224	. 2 ⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ↔ (𝜑 → 𝜓))
3		nanbi 1500	. 2 ⊢ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ↔ (𝜑 → 𝜓)) ↔ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 → 𝜓)) ⊼ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 ⊼ (𝜓 ⊼ 𝜓))) ⊼ ((𝜑 → 𝜓) ⊼ (𝜑 → 𝜓)))))
4	2, 3	mpbi 230	1 ⊢ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 → 𝜓)) ⊼ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 ⊼ (𝜓 ⊼ 𝜓))) ⊼ ((𝜑 → 𝜓) ⊼ (𝜑 → 𝜓))))

Syntax hints:   → wi 4   ↔ wb 206   ⊼ wnan 1491

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-an 396  df-or 849  df-nan 1492

This theorem is referenced by:  nic-stdmp  1690  nic-luk1  1691  nic-luk2  1692  nic-luk3  1693