Theorem 2false 378

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 267	. 2 ⊢ (¬ 𝜑 ↔ ¬ 𝜓)
4	3	con4bii 324	1 ⊢ (𝜑 ↔ 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by:  bianfi  543  bifal  1586  dfnul3  4293  co02  6267  0er  8742  00lss  21099  00ply1bas  22436  2lgslem4  27607  signswch  34980  pexmidlem8N  40792  dandysum2p2e4  47776