Theorem notnotr 130

Description: Double negation elimination. Converse of notnot 142 and one implication of notnotb 316. 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  316  necon1ad  2951  necon4bd  2954  noetasuplem4  27718  eulercrct  30330  expgt0b  32909  notornotel1  38462  mpobi123f  38529  mptbi12f  38533  oexpreposd  42799  axfrege31  44277  clsk1independent  44490  con3ALT2  44974  zfregs2VD  45284  con3ALTVD  45359  notnotrALT2  45370  suplesup  45784