Theorem 2false 690

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 635	. 2 ⊢ (𝜑 → 𝜓)
3		2false.2	. . 3 ⊢ ¬ 𝜓
4	3	pm2.21i 635	. 2 ⊢ (𝜓 → 𝜑)
5	2, 4	impbii 125	1 ⊢ (𝜑 ↔ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 106  ax-ia3 107  ax-in2 604

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bianfi  931  bifal  1344  dfnul2  3360  dfnul3  3361  rab0  3386  iun0  3864  0iun  3865  0xp  4614  cnv0  4937  co02  5047  0er  6456  bdnth  13021  bdnthALT  13022