MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nic-dfim Structured version   Visualization version   GIF version

Theorem nic-dfim 1584
Description: Define implication in terms of 'nand'. Analogous to ((𝜑 ⊼ (𝜓𝜓)) ↔ (𝜑𝜓)). 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 1443 . . 3 ((𝜑𝜓) ↔ (𝜑 ⊼ (𝜓𝜓)))
21bicomi 212 . 2 ((𝜑 ⊼ (𝜓𝜓)) ↔ (𝜑𝜓))
3 nanbi 1445 . 2 (((𝜑 ⊼ (𝜓𝜓)) ↔ (𝜑𝜓)) ↔ (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑𝜓)) ⊼ (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑 ⊼ (𝜓𝜓))) ⊼ ((𝜑𝜓) ⊼ (𝜑𝜓)))))
42, 3mpbi 218 1 (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑𝜓)) ⊼ (((𝜑 ⊼ (𝜓𝜓)) ⊼ (𝜑 ⊼ (𝜓𝜓))) ⊼ ((𝜑𝜓) ⊼ (𝜑𝜓))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wnan 1438
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 195  df-or 383  df-an 384  df-nan 1439
This theorem is referenced by:  nic-stdmp  1605  nic-luk1  1606  nic-luk2  1607  nic-luk3  1608
  Copyright terms: Public domain W3C validator