Theorem 2false 649

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 607	. 2 ⊢ (𝜑 → 𝜓)
3		2false.2	. . 3 ⊢ ¬ 𝜓
4	3	pm2.21i 607	. 2 ⊢ (𝜓 → 𝜑)
5	2, 4	impbii 124	1 ⊢ (𝜑 ↔ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia2 105  ax-ia3 106  ax-in2 577

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  bianfi  888  bifal  1297  dfnul2  3253  dfnul3  3254  rab0  3273  iun0  3734  0iun  3735  0xp  4438  cnv0  4747  co02  4854  0er  6163  bdnth  10625  bdnthALT  10626