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Theorem nic-ich 1677
Description: Chained inference. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
nic-ich.1 (𝜑 ⊼ (𝜓𝜓))
nic-ich.2 (𝜓 ⊼ (𝜒𝜒))
Assertion
Ref Expression
nic-ich (𝜑 ⊼ (𝜒𝜒))

Proof of Theorem nic-ich
StepHypRef Expression
1 nic-ich.2 . . 3 (𝜓 ⊼ (𝜒𝜒))
21nic-isw1 1672 . 2 ((𝜒𝜒) ⊼ 𝜓)
3 nic-ich.1 . . 3 (𝜑 ⊼ (𝜓𝜓))
43nic-imp 1667 . 2 (((𝜒𝜒) ⊼ 𝜓) ⊼ ((𝜑 ⊼ (𝜒𝜒)) ⊼ (𝜑 ⊼ (𝜒𝜒))))
52, 4nic-mp 1663 1 (𝜑 ⊼ (𝜒𝜒))
Colors of variables: wff setvar class
Syntax hints:  wnan 1475
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 208  df-an 397  df-nan 1476
This theorem is referenced by:  nic-idbl  1678  nic-luk1  1683
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