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Theorem nic-imp 1590
Description: Inference for nic-mp 1586 using nic-ax 1588 as major premise. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
nic-imp.1 (𝜑 ⊼ (𝜒𝜓))
Assertion
Ref Expression
nic-imp ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))

Proof of Theorem nic-imp
StepHypRef Expression
1 nic-imp.1 . 2 (𝜑 ⊼ (𝜒𝜓))
2 nic-ax 1588 . 2 ((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
31, 2nic-mp 1586 1 ((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))
Colors of variables: wff setvar class
Syntax hints:  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-an 384  df-nan 1439
This theorem is referenced by:  nic-idlem1  1591  nic-idlem2  1592  nic-isw2  1596  nic-iimp1  1597  nic-idel  1599  nic-ich  1600  nic-idbl  1601  nic-luk1  1606
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