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  2950  necon4bd  2953  noetasuplem4  27716  eulercrct  30329  expgt0b  32908  notornotel1  38346  mpobi123f  38413  mptbi12f  38417  oexpreposd  42692  axfrege31  44189  clsk1independent  44402  con3ALT2  44886  zfregs2VD  45196  con3ALTVD  45271  notnotrALT2  45282  suplesup  45698