MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2false Structured version   Visualization version   GIF version

Theorem 2false 367
Description: Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
Hypotheses
Ref Expression
2false.1 ¬ 𝜑
2false.2 ¬ 𝜓
Assertion
Ref Expression
2false (𝜑𝜓)

Proof of Theorem 2false
StepHypRef Expression
1 2false.1 . . 3 ¬ 𝜑
2 2false.2 . . 3 ¬ 𝜓
31, 22th 256 . 2 𝜑 ↔ ¬ 𝜓)
43con4bii 313 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198
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 199
This theorem is referenced by:  bianfi  529  bifal  1618  dfnul2  4143  co02  5905  0er  8065  00lss  19338  00ply1bas  20010  2lgslem4  25587  signswch  31242  pexmidlem8N  36136  dandysum2p2e4  42102
  Copyright terms: Public domain W3C validator