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

Theorem notnotr 130
Description: Double negation elimination. Converse of notnot 142 and one implication of notnotb 314. Theorem *2.14 of [WhiteheadRussell] p. 102. This was the fifth axiom of Frege, specifically Proposition 31 of [Frege1879] p. 44. In classical logic (our logic) this is always true. In intuitionistic logic this is not always true, and formulas for which it is true are called "stable". (Contributed by NM, 29-Dec-1992.) (Proof shortened by David Harvey, 5-Sep-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.)
Assertion
Ref Expression
notnotr (¬ ¬ 𝜑𝜑)

Proof of Theorem notnotr
StepHypRef Expression
1 pm2.18 128 . 2 ((¬ 𝜑𝜑) → 𝜑)
21jarli 126 1 (¬ ¬ 𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem is referenced by:  notnotrd  133  con2d  134  con3d  152  notnotb  314  ecase3adOLD  1034  necon1ad  2954  necon4bd  2957  noetasuplem4  27689  eulercrct  30072  notornotel1  37601  mpobi123f  37668  mptbi12f  37672  oexpreposd  41912  axfrege31  43294  clsk1independent  43507  con3ALT2  44000  zfregs2VD  44311  con3ALTVD  44386  notnotrALT2  44397  suplesup  44750
  Copyright terms: Public domain W3C validator