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	⊢ (¬ ¬ 𝜑 → 𝜑)

Proof of Theorem notnotr

Step	Hyp	Ref	Expression
1		pm2.18 128	. 2 ⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
2	1	jarli 126	1 ⊢ (¬ ¬ 𝜑 → 𝜑)

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  2942  necon4bd  2945  noetasuplem4  27646  eulercrct  30186  expgt0b  32761  notornotel1  38079  mpobi123f  38146  mptbi12f  38150  oexpreposd  42299  axfrege31  43810  clsk1independent  44023  con3ALT2  44508  zfregs2VD  44818  con3ALTVD  44893  notnotrALT2  44904  suplesup  45323