Theorem notnotr 132

Description: Double negation elimination. Converse of notnot 144 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		pm2.18 128	. 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  135  con2d  136  con3d  155  notnotb  318  ecase3ad  1032  necon1ad  3004  necon4bd  3007  eulercrct  28027  noetalem3  33332  notornotel1  35533  mpobi123f  35600  mptbi12f  35604  oexpreposd  39487  axfrege31  40534  clsk1independent  40749  con3ALT2  41236  zfregs2VD  41547  con3ALTVD  41622  notnotrALT2  41633  suplesup  41971