Theorem falim 1587

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 1580. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)

Assertion

Ref	Expression
falim	⊢ (⊥ → 𝜑)

Proof of Theorem falim

Step	Hyp	Ref	Expression
1		fal 1584	. 2 ⊢ ¬ ⊥
2	1	pm2.21i 120	1 ⊢ (⊥ → 𝜑)

This proof depends on syntax axioms:   → wi 4  ⊥wfal 1582

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-tru 1573  df-fal 1583

This theorem is used by:  falimd  1588  tbw-bijust  1731  tbw-negdf  1732  tbw-ax4  1736  merco1  1746  merco2  1769  csbprc  4374  ralnralall  4476  tgcgr4  28851  frgrregord013  30817  nalfal  36971  imsym1  36986  consym1  36988  dissym1  36989  unisym1  36991  exisym1  36992  subsym1  36995  bj-falor2  37235  bj-cbvaw  37320  bj-cbveaw  37322  bj-prmoore  37814  wl-2mintru2  38194  orfa1  38794  orfa2  38795  bifald  38796  botel  38811  quadfac  43030  quantgodelALT  47647  lindslinindsimp2  49300