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Theorem nic-dfneg 1742
 Description: Define negation in terms of 'nand'. Analogous to ((𝜑 ⊼ 𝜑) ↔ ¬ 𝜑). 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-dfneg (((𝜑𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑𝜑) ⊼ (𝜑𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑)))

Proof of Theorem nic-dfneg
StepHypRef Expression
1 nannot 1600 . . 3 𝜑 ↔ (𝜑𝜑))
21bicomi 214 . 2 ((𝜑𝜑) ↔ ¬ 𝜑)
3 nanbi 1601 . 2 (((𝜑𝜑) ↔ ¬ 𝜑) ↔ (((𝜑𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑𝜑) ⊼ (𝜑𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑))))
42, 3mpbi 220 1 (((𝜑𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑𝜑) ⊼ (𝜑𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ⊼ wnan 1594 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 197  df-an 383  df-or 827  df-nan 1595 This theorem is referenced by:  nic-luk2  1764  nic-luk3  1765
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