Theorem bifal 1550

Description: A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)

Hypothesis

Ref	Expression
bifal.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
bifal	⊢ (𝜑 ↔ ⊥)

Proof of Theorem bifal

Step	Hyp	Ref	Expression
1		bifal.1	. 2 ⊢ ¬ 𝜑
2		fal 1548	. 2 ⊢ ¬ ⊥
3	1, 2	2false 374	1 ⊢ (𝜑 ↔ ⊥)

Syntax hints:  ¬ wn 3   ↔ wb 205  ⊥wfal 1546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-tru 1537  df-fal 1547

This theorem is referenced by:  falantru  1569  dfnul4  4337  dfnul2  4338  dfnul4OLD  4342  abf  4413  ralnralall  4526  tgcgr4  28481  frgrregord013  30351  nrmo  36170  bj-ntrufal  36321  bicontr  37828  aibnbaif  46592  aifftbifffaibif  46606  atnaiana  46608