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Theorem nic-dfneg 1662
Description: This theorem "defines" negation in terms of 'nand'. Analogous to nannot 1483. 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 1483 . . 3 𝜑 ↔ (𝜑𝜑))
21bicomi 225 . 2 ((𝜑𝜑) ↔ ¬ 𝜑)
3 nanbi 1484 . 2 (((𝜑𝜑) ↔ ¬ 𝜑) ↔ (((𝜑𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑𝜑) ⊼ (𝜑𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑))))
42, 3mpbi 231 1 (((𝜑𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑𝜑) ⊼ (𝜑𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wnan 1475
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 208  df-an 397  df-or 842  df-nan 1476
This theorem is referenced by:  nic-luk2  1684  nic-luk3  1685
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