Theorem 2false 701

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 646	. 2 ⊢ (𝜑 → 𝜓)
3		2false.2	. . 3 ⊢ ¬ 𝜓
4	3	pm2.21i 646	. 2 ⊢ (𝜓 → 𝜑)
5	2, 4	impbii 126	1 ⊢ (𝜑 ↔ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108  ax-in2 615

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bianfi  947  bifal  1366  dfnul2  3426  dfnul3  3427  rab0  3453  iun0  3945  0iun  3946  0xp  4708  cnv0  5034  co02  5144  0er  6571  bdnth  14671  bdnthALT  14672