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Theorem notnotr 130
Description: Double negation elimination. Converse of notnot 142 and one implication of notnotb 315. 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  315  necon1ad  2956  necon4bd  2959  noetasuplem4  27771  eulercrct  30251  expgt0b  32806  notornotel1  38080  mpobi123f  38147  mptbi12f  38151  oexpreposd  42335  axfrege31  43824  clsk1independent  44037  con3ALT2  44528  zfregs2VD  44839  con3ALTVD  44914  notnotrALT2  44925  suplesup  45323
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