Theorem notnot 143

Description: Double negation introduction. Converse of notnotr 131 and one implication of notnotb 318. Theorem *2.12 of [WhiteheadRussell] p. 101. This was the sixth axiom of Frege, specifically Proposition 41 of [Frege1879] p. 47. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)

Assertion

Ref	Expression
notnot	⊢ (𝜑 → ¬ ¬ 𝜑)

Proof of Theorem notnot

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ (¬ 𝜑 → ¬ 𝜑)
2	1	con2i 140	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:  notnoti  144  notnotd  145  con1d  146  notnotb  318  pm2.13  911  biortn  951  necon2ad  2970  necon4ad  2974  necon4ai  2986  eueq2  3668  ifnot  4535  spthcycl  30272  knoppndvlem10  37219  wl-orel12  38275  cnfn1dd  38841  cnfn2dd  38842  axfrege41  44685  vk15.4j  45352  zfregs2VD  45664  vk15.4jVD  45737  con3ALTVD  45739  stoweidlem39  46868