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)

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  134  con2d  135  con3d  153  notnotb  318  necon1ad  2975  necon4bd  2978  noetasuplem4  27900  eulercrct  30593  expgt0b  33161  notornotel1  38764  mpobi123f  38831  mptbi12f  38835  oexpreposd  43103  axfrege31  44579  clsk1independent  44792  con3ALT2  45259  zfregs2VD  45569  con3ALTVD  45644  notnotrALT2  45655  suplesup  46075