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  2976  necon4bd  2979  noetasuplem4  28150  eulercrct  30900  expgt0b  33504  notornotel1  39293  mpobi123f  39360  mptbi12f  39364  oexpreposd  43645  axfrege31  45094  clsk1independent  45307  con3ALT2  45773  zfregs2VD  46083  con3ALTVD  46158  notnotrALT2  46169  suplesup  46624