Theorem 2false 713

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 655	. 2 ⊢ (𝜑 → 𝜓)
3		2false.2	. . 3 ⊢ ¬ 𝜓
4	3	pm2.21i 655	. 2 ⊢ (𝜓 → 𝜑)
5	2, 4	impbii 126	1 ⊢ (𝜑 ↔ 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108  ax-in2 624

This proof depends on definitions:  df-bi 117

This theorem is used by:  bianfi  960  bifal  1415  dfnul3  3524  rab0  3551  iun0  4069  0iun  4070  0xp  4855  cnv0  5191  co02  5301  0er  6841  2lgslem4  16234  bdnth  16872  bdnthALT  16873