MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nic-idlem2 Structured version   Visualization version   GIF version

Theorem nic-idlem2 1751
Description: Lemma for nic-id 1752. Inference used by nic-id 1752. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
nic-idlem2.1 (𝜂 ⊼ ((𝜑 ⊼ (𝜒𝜓)) ⊼ 𝜃))
Assertion
Ref Expression
nic-idlem2 ((𝜃 ⊼ (𝜏 ⊼ (𝜏𝜏))) ⊼ 𝜂)

Proof of Theorem nic-idlem2
StepHypRef Expression
1 nic-idlem2.1 . 2 (𝜂 ⊼ ((𝜑 ⊼ (𝜒𝜓)) ⊼ 𝜃))
2 nic-ax 1747 . . . 4 ((𝜑 ⊼ (𝜒𝜓)) ⊼ ((𝜏 ⊼ (𝜏𝜏)) ⊼ ((𝜑𝜒) ⊼ ((𝜑𝜑) ⊼ (𝜑𝜑)))))
32nic-imp 1749 . . 3 ((𝜃 ⊼ (𝜏 ⊼ (𝜏𝜏))) ⊼ (((𝜑 ⊼ (𝜒𝜓)) ⊼ 𝜃) ⊼ ((𝜑 ⊼ (𝜒𝜓)) ⊼ 𝜃)))
43nic-imp 1749 . 2 ((𝜂 ⊼ ((𝜑 ⊼ (𝜒𝜓)) ⊼ 𝜃)) ⊼ (((𝜃 ⊼ (𝜏 ⊼ (𝜏𝜏))) ⊼ 𝜂) ⊼ ((𝜃 ⊼ (𝜏 ⊼ (𝜏𝜏))) ⊼ 𝜂)))
51, 4nic-mp 1745 1 ((𝜃 ⊼ (𝜏 ⊼ (𝜏𝜏))) ⊼ 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wnan 1596
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 385  df-nan 1597
This theorem is referenced by:  nic-id  1752
  Copyright terms: Public domain W3C validator