MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm5.16 Structured version   Visualization version   GIF version

Theorem pm5.16 1013
Description: Theorem *5.16 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 17-Oct-2013.)
Assertion
Ref Expression
pm5.16 ¬ ((𝜑𝜓) ∧ (𝜑 ↔ ¬ 𝜓))

Proof of Theorem pm5.16
StepHypRef Expression
1 pm5.18 386 . . 3 ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
21biimpi 219 . 2 ((𝜑𝜓) → ¬ (𝜑 ↔ ¬ 𝜓))
3 imnan 403 . 2 (((𝜑𝜓) → ¬ (𝜑 ↔ ¬ 𝜓)) ↔ ¬ ((𝜑𝜓) ∧ (𝜑 ↔ ¬ 𝜓)))
42, 3mpbi 233 1 ¬ ((𝜑𝜓) ∧ (𝜑 ↔ ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399
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 210  df-an 400
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator