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Theorem nic-dfim 1713
 Description: This theorem "defines" implication in terms of 'nand'. Analogous to nanim 1566. 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 1566 . . 3 ((𝜑𝜓) ↔ (𝜑 ⊼ (𝜓𝜓)))
21bicomi 216 . 2 ((𝜑 ⊼ (𝜓𝜓)) ↔ (𝜑𝜓))
3 nanbi 1569 . 2 (((𝜑 ⊼ (𝜓𝜓)) ↔ (𝜑𝜓)) ↔ (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑𝜓)) ⊼ (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑 ⊼ (𝜓𝜓))) ⊼ ((𝜑𝜓) ⊼ (𝜑𝜓)))))
42, 3mpbi 222 1 (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑𝜓)) ⊼ (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑 ⊼ (𝜓𝜓))) ⊼ ((𝜑𝜓) ⊼ (𝜑𝜓))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ⊼ wnan 1557 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 199  df-an 387  df-or 837  df-nan 1558 This theorem is referenced by:  nic-stdmp  1734  nic-luk1  1735  nic-luk2  1736  nic-luk3  1737
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