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Theorem notnotr 132
Description: Double negation elimination. Converse of notnot 144 and one implication of notnotb 317. 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  135  con2d  136  con3d  155  notnotb  317  ecase3ad  1030  necon1ad  3031  necon4bd  3034  eulercrct  28013  noetalem3  33207  notornotel1  35360  mpobi123f  35427  mptbi12f  35431  oexpreposd  39164  axfrege31  40164  clsk1independent  40381  con3ALT2  40849  zfregs2VD  41160  con3ALTVD  41235  notnotrALT2  41246  suplesup  41591
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