Theorem notnot 282

Description: Double negation. Theorem *4.13 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
notnot	⊢ (φ ↔ ¬ ¬ φ)

Proof of Theorem notnot

Step	Hyp	Ref	Expression
1		notnot1 114	. 2 ⊢ (φ → ¬ ¬ φ)
2		notnot2 104	. 2 ⊢ (¬ ¬ φ → φ)
3	1, 2	impbii 180	1 ⊢ (φ ↔ ¬ ¬ φ)

Syntax hints:  ¬ wn 3   ↔ wb 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  notbid  285  con2bi  318  con1bii  321  imor  401  iman  413  dfbi3  863  alex  1572  19.8wOLD  1693  sbn  2062  difsnpss  3852  dfimak2  4299