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  2945  necon4bd  2948  noetasuplem4  27675  eulercrct  30222  expgt0b  32799  notornotel1  38145  mpobi123f  38212  mptbi12f  38216  oexpreposd  42425  axfrege31  43936  clsk1independent  44149  con3ALT2  44633  zfregs2VD  44943  con3ALTVD  45018  notnotrALT2  45029  suplesup  45448