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

Theorem nic-mp 1745
Description: Derive Nicod's rule of modus ponens using 'nand', from the standard one. Although the major and minor premise together also imply 𝜒, this form is necessary for useful derivations from nic-ax 1747. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom ($a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
nic-jmin 𝜑
nic-jmaj (𝜑 ⊼ (𝜒𝜓))
Assertion
Ref Expression
nic-mp 𝜓

Proof of Theorem nic-mp
StepHypRef Expression
1 nic-jmin . 2 𝜑
2 nic-jmaj . . . 4 (𝜑 ⊼ (𝜒𝜓))
3 nannan 1600 . . . 4 ((𝜑 ⊼ (𝜒𝜓)) ↔ (𝜑 → (𝜒𝜓)))
42, 3mpbi 220 . . 3 (𝜑 → (𝜒𝜓))
54simprd 482 . 2 (𝜑𝜓)
61, 5ax-mp 5 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wnan 1596
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 385  df-nan 1597
This theorem is referenced by:  nic-imp  1749  nic-idlem2  1751  nic-id  1752  nic-swap  1753  nic-isw1  1754  nic-isw2  1755  nic-iimp1  1756  nic-idel  1758  nic-ich  1759  nic-stdmp  1764  nic-luk1  1765  nic-luk2  1766  nic-luk3  1767  lukshefth1  1769  lukshefth2  1770  renicax  1771
  Copyright terms: Public domain W3C validator