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

Theorem nic-idbl 1760
Description: Double the terms. Since doubling is the same as negation, this can be viewed as a contraposition inference. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
nic-idbl.1 (𝜑 ⊼ (𝜓𝜓))
Assertion
Ref Expression
nic-idbl ((𝜓𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜑𝜑)))

Proof of Theorem nic-idbl
StepHypRef Expression
1 nic-idbl.1 . . 3 (𝜑 ⊼ (𝜓𝜓))
21nic-imp 1749 . 2 ((𝜓𝜓) ⊼ ((𝜑𝜓) ⊼ (𝜑𝜓)))
31nic-imp 1749 . 2 ((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜑𝜑)))
42, 3nic-ich 1759 1 ((𝜓𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜑𝜑)))
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-luk1  1765
  Copyright terms: Public domain W3C validator