Theorem 2false 367

Description: Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)

Hypotheses

Ref	Expression
2false.1	⊢ ¬ 𝜑
2false.2	⊢ ¬ 𝜓

Assertion

Ref	Expression
2false	⊢ (𝜑 ↔ 𝜓)

Proof of Theorem 2false

Step	Hyp	Ref	Expression
1		2false.1	. . 3 ⊢ ¬ 𝜑
2		2false.2	. . 3 ⊢ ¬ 𝜓
3	1, 2	2th 256	. 2 ⊢ (¬ 𝜑 ↔ ¬ 𝜓)
4	3	con4bii 313	1 ⊢ (𝜑 ↔ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 198

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 199

This theorem is referenced by:  bianfi  529  bifal  1618  dfnul2  4143  co02  5905  0er  8065  00lss  19338  00ply1bas  20010  2lgslem4  25587  signswch  31242  pexmidlem8N  36136  dandysum2p2e4  42102