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Theorem nic-luk2 1764
Description: Proof of luk-2 1728 from nic-ax 1745 and nic-mp 1743. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nic-luk2 ((¬ 𝜑𝜑) → 𝜑)

Proof of Theorem nic-luk2
StepHypRef Expression
1 nic-dfim 1741 . . . . 5 (((¬ 𝜑 ⊼ (𝜑𝜑)) ⊼ (¬ 𝜑𝜑)) ⊼ (((¬ 𝜑 ⊼ (𝜑𝜑)) ⊼ (¬ 𝜑 ⊼ (𝜑𝜑))) ⊼ ((¬ 𝜑𝜑) ⊼ (¬ 𝜑𝜑))))
21nic-bi2 1761 . . . 4 ((¬ 𝜑𝜑) ⊼ ((¬ 𝜑 ⊼ (𝜑𝜑)) ⊼ (¬ 𝜑 ⊼ (𝜑𝜑))))
3 nic-dfneg 1742 . . . . . 6 (((𝜑𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑𝜑) ⊼ (𝜑𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑)))
4 nic-id 1750 . . . . . 6 ((𝜑𝜑) ⊼ ((𝜑𝜑) ⊼ (𝜑𝜑)))
53, 4nic-iimp1 1754 . . . . 5 ((𝜑𝜑) ⊼ ((𝜑𝜑) ⊼ ¬ 𝜑))
65nic-isw2 1753 . . . 4 ((𝜑𝜑) ⊼ (¬ 𝜑 ⊼ (𝜑𝜑)))
72, 6nic-iimp1 1754 . . 3 ((𝜑𝜑) ⊼ (¬ 𝜑𝜑))
87nic-isw1 1752 . 2 ((¬ 𝜑𝜑) ⊼ (𝜑𝜑))
9 nic-dfim 1741 . . 3 ((((¬ 𝜑𝜑) ⊼ (𝜑𝜑)) ⊼ ((¬ 𝜑𝜑) → 𝜑)) ⊼ ((((¬ 𝜑𝜑) ⊼ (𝜑𝜑)) ⊼ ((¬ 𝜑𝜑) ⊼ (𝜑𝜑))) ⊼ (((¬ 𝜑𝜑) → 𝜑) ⊼ ((¬ 𝜑𝜑) → 𝜑))))
109nic-bi1 1760 . 2 (((¬ 𝜑𝜑) ⊼ (𝜑𝜑)) ⊼ (((¬ 𝜑𝜑) → 𝜑) ⊼ ((¬ 𝜑𝜑) → 𝜑)))
118, 10nic-mp 1743 1 ((¬ 𝜑𝜑) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wnan 1594
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 827  df-nan 1595
This theorem is referenced by: (None)
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