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Theorem nic-dfim 1585
 Description: Define implication in terms of 'nand'. Analogous to ((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ↔ (𝜑 → 𝜓)). 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 ⊼ (𝜓𝜓)) ⊼ (𝜑𝜓)) ⊼ (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑 ⊼ (𝜓𝜓))) ⊼ ((𝜑𝜓) ⊼ (𝜑𝜓))))

Proof of Theorem nic-dfim
StepHypRef Expression
1 nanim 1444 . . 3 ((𝜑𝜓) ↔ (𝜑 ⊼ (𝜓𝜓)))
21bicomi 213 . 2 ((𝜑 ⊼ (𝜓𝜓)) ↔ (𝜑𝜓))
3 nanbi 1446 . 2 (((𝜑 ⊼ (𝜓𝜓)) ↔ (𝜑𝜓)) ↔ (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑𝜓)) ⊼ (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑 ⊼ (𝜓𝜓))) ⊼ ((𝜑𝜓) ⊼ (𝜑𝜓)))))
42, 3mpbi 219 1 (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑𝜓)) ⊼ (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑 ⊼ (𝜓𝜓))) ⊼ ((𝜑𝜓) ⊼ (𝜑𝜓))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ⊼ wnan 1439 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 196  df-or 384  df-an 385  df-nan 1440 This theorem is referenced by:  nic-stdmp  1606  nic-luk1  1607  nic-luk2  1608  nic-luk3  1609
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