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  2975  necon4bd  2978  noetasuplem4  28105  eulercrct  30855  expgt0b  33461  notornotel1  39167  mpobi123f  39234  mptbi12f  39238  oexpreposd  43519  axfrege31  44973  clsk1independent  45186  con3ALT2  45652  zfregs2VD  45962  con3ALTVD  46037  notnotrALT2  46048  suplesup  46490