Theorem notnotr 131

Description: Double negation elimination. Converse of notnot 143 and one implication of notnotb 318. 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		pm2.18 129	. 2 ⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
2	1	jarli 127	1 ⊢ (¬ ¬ 𝜑 → 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

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

This theorem is used by:  notnotrd  134  con2d  135  con3d  153  notnotb  318  necon1ad  2977  necon4bd  2980  noetasuplem4  27953  eulercrct  30666  expgt0b  33233  notornotel1  38804  mpobi123f  38871  mptbi12f  38875  oexpreposd  43143  axfrege31  44619  clsk1independent  44832  con3ALT2  45299  zfregs2VD  45609  con3ALTVD  45684  notnotrALT2  45695  suplesup  46115