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

Theorem falim 1555
Description: The truth value implies anything. Also called the "principle of explosion", or "ex falso [sequitur]] quodlibet" (Latin for "from falsehood, anything [follows]]"). Dual statement of trud 1548. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
Assertion
Ref Expression
falim (⊥ → 𝜑)

Proof of Theorem falim
StepHypRef Expression
1 fal 1552 . 2 ¬ ⊥
21pm2.21i 119 1 (⊥ → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wfal 1550
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-tru 1541  df-fal 1551
This theorem is referenced by:  falimd  1556  tbw-bijust  1700  tbw-negdf  1701  tbw-ax4  1705  merco1  1715  merco2  1738  csbprc  4340  ralnralall  4440  tgcgr4  26321  frgrregord013  28176  nalfal  33776  imsym1  33791  consym1  33793  dissym1  33794  unisym1  33796  exisym1  33797  bj-falor2  33944  bj-prmoore  34443  wl-2mintru2  34819  orfa1  35433  orfa2  35434  bifald  35435  botel  35452  lindslinindsimp2  44734
  Copyright terms: Public domain W3C validator