Theorem 2false 374

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 263	. 2 ⊢ (¬ 𝜑 ↔ ¬ 𝜓)
4	3	con4bii 320	1 ⊢ (𝜑 ↔ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 205

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 206

This theorem is referenced by:  bianfi  532  bifal  1549  dfnul3  4330  dfnul2OLD  4331  co02  6269  0er  8768  00lss  20832  00ply1bas  22165  2lgslem4  27359  signswch  34226  pexmidlem8N  39482  dandysum2p2e4  46409