Theorem 2false 365

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 254	. 2 ⊢ (¬ 𝜑 ↔ ¬ 𝜓)
4	3	con4bii 311	1 ⊢ (𝜑 ↔ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 196

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 197

This theorem is referenced by:  bianfi  966  bifal  1497  cnv0OLD  5536  co02  5649  0er  7780  0erOLD  7781  00lss  18942  00ply1bas  19610  2lgslem4  25131  signswch  30638  pexmidlem8N  35263  dandysum2p2e4  41165