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Theorem nic-luk3 1609
 Description: Proof of luk-3 1573 from nic-ax 1589 and nic-mp 1587. (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 1585 . . . 4 (((¬ 𝜑 ⊼ (𝜓𝜓)) ⊼ (¬ 𝜑𝜓)) ⊼ (((¬ 𝜑 ⊼ (𝜓𝜓)) ⊼ (¬ 𝜑 ⊼ (𝜓𝜓))) ⊼ ((¬ 𝜑𝜓) ⊼ (¬ 𝜑𝜓))))
21nic-bi1 1604 . . 3 ((¬ 𝜑 ⊼ (𝜓𝜓)) ⊼ ((¬ 𝜑𝜓) ⊼ (¬ 𝜑𝜓)))
3 nic-dfneg 1586 . . . . 5 (((𝜑𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑𝜑) ⊼ (𝜑𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑)))
43nic-bi2 1605 . . . 4 𝜑 ⊼ ((𝜑𝜑) ⊼ (𝜑𝜑)))
5 nic-id 1594 . . . 4 (𝜑 ⊼ (𝜑𝜑))
64, 5nic-iimp1 1598 . . 3 (𝜑 ⊼ ¬ 𝜑)
72, 6nic-iimp2 1599 . 2 (𝜑 ⊼ ((¬ 𝜑𝜓) ⊼ (¬ 𝜑𝜓)))
8 nic-dfim 1585 . . 3 (((𝜑 ⊼ ((¬ 𝜑𝜓) ⊼ (¬ 𝜑𝜓))) ⊼ (𝜑 → (¬ 𝜑𝜓))) ⊼ (((𝜑 ⊼ ((¬ 𝜑𝜓) ⊼ (¬ 𝜑𝜓))) ⊼ (𝜑 ⊼ ((¬ 𝜑𝜓) ⊼ (¬ 𝜑𝜓)))) ⊼ ((𝜑 → (¬ 𝜑𝜓)) ⊼ (𝜑 → (¬ 𝜑𝜓)))))
98nic-bi1 1604 . 2 ((𝜑 ⊼ ((¬ 𝜑𝜓) ⊼ (¬ 𝜑𝜓))) ⊼ ((𝜑 → (¬ 𝜑𝜓)) ⊼ (𝜑 → (¬ 𝜑𝜓))))
107, 9nic-mp 1587 1 (𝜑 → (¬ 𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ⊼ wnan 1439 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 196  df-or 384  df-an 385  df-nan 1440 This theorem is referenced by: (None)
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