Theorem 2false 617

Description: Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)

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	1	pm2.21i 575	. 2 ⊢ (𝜑 → 𝜓)
3		2false.2	. . 3 ⊢ ¬ 𝜓
4	3	pm2.21i 575	. 2 ⊢ (𝜓 → 𝜑)
5	2, 4	impbii 117	1 ⊢ (𝜑 ↔ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia2 100  ax-ia3 101  ax-in2 545

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  bianfi  854  bifal  1256  dfnul2  3226  dfnul3  3227  rab0  3246  iun0  3713  0iun  3714  0xp  4420  cnv0  4727  co02  4834  0er  6140  bdnth  9954  bdnthALT  9955