MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nic-luk3 Structured version   Visualization version   GIF version

Theorem nic-luk3 1766
Description: Proof of luk-3 1730 from nic-ax 1746 and nic-mp 1744. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nic-luk3 (𝜑 → (¬ 𝜑𝜓))

Proof of Theorem nic-luk3
StepHypRef Expression
1 nic-dfim 1742 . . . 4 (((¬ 𝜑 ⊼ (𝜓𝜓)) ⊼ (¬ 𝜑𝜓)) ⊼ (((¬ 𝜑 ⊼ (𝜓𝜓)) ⊼ (¬ 𝜑 ⊼ (𝜓𝜓))) ⊼ ((¬ 𝜑𝜓) ⊼ (¬ 𝜑𝜓))))
21nic-bi1 1761 . . 3 ((¬ 𝜑 ⊼ (𝜓𝜓)) ⊼ ((¬ 𝜑𝜓) ⊼ (¬ 𝜑𝜓)))
3 nic-dfneg 1743 . . . . 5 (((𝜑𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑𝜑) ⊼ (𝜑𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑)))
43nic-bi2 1762 . . . 4 𝜑 ⊼ ((𝜑𝜑) ⊼ (𝜑𝜑)))
5 nic-id 1751 . . . 4 (𝜑 ⊼ (𝜑𝜑))
64, 5nic-iimp1 1755 . . 3 (𝜑 ⊼ ¬ 𝜑)
72, 6nic-iimp2 1756 . 2 (𝜑 ⊼ ((¬ 𝜑𝜓) ⊼ (¬ 𝜑𝜓)))
8 nic-dfim 1742 . . 3 (((𝜑 ⊼ ((¬ 𝜑𝜓) ⊼ (¬ 𝜑𝜓))) ⊼ (𝜑 → (¬ 𝜑𝜓))) ⊼ (((𝜑 ⊼ ((¬ 𝜑𝜓) ⊼ (¬ 𝜑𝜓))) ⊼ (𝜑 ⊼ ((¬ 𝜑𝜓) ⊼ (¬ 𝜑𝜓)))) ⊼ ((𝜑 → (¬ 𝜑𝜓)) ⊼ (𝜑 → (¬ 𝜑𝜓)))))
98nic-bi1 1761 . 2 ((𝜑 ⊼ ((¬ 𝜑𝜓) ⊼ (¬ 𝜑𝜓))) ⊼ ((𝜑 → (¬ 𝜑𝜓)) ⊼ (𝜑 → (¬ 𝜑𝜓))))
107, 9nic-mp 1744 1 (𝜑 → (¬ 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wnan 1595
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 837  df-nan 1596
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator