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)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by:  notnotrd  134  con2d  135  con3d  153  notnotb  318  necon1ad  2972  necon4bd  2975  noetasuplem4  27973  eulercrct  30723  expgt0b  33288  notornotel1  38844  mpobi123f  38911  mptbi12f  38915  oexpreposd  43198  axfrege31  44674  clsk1independent  44887  con3ALT2  45354  zfregs2VD  45664  con3ALTVD  45739  notnotrALT2  45750  suplesup  46170