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Theorem nic-dfim 1661
Description: This theorem "defines" implication in terms of 'nand'. Analogous to nanim 1482. 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 1482 . . 3 ((𝜑𝜓) ↔ (𝜑 ⊼ (𝜓𝜓)))
21bicomi 225 . 2 ((𝜑 ⊼ (𝜓𝜓)) ↔ (𝜑𝜓))
3 nanbi 1484 . 2 (((𝜑 ⊼ (𝜓𝜓)) ↔ (𝜑𝜓)) ↔ (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑𝜓)) ⊼ (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑 ⊼ (𝜓𝜓))) ⊼ ((𝜑𝜓) ⊼ (𝜑𝜓)))))
42, 3mpbi 231 1 (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑𝜓)) ⊼ (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑 ⊼ (𝜓𝜓))) ⊼ ((𝜑𝜓) ⊼ (𝜑𝜓))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  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-stdmp  1682  nic-luk1  1683  nic-luk2  1684  nic-luk3  1685
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