Theorem bifal 1538

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 1536	. 2 ⊢ ¬ ⊥
3	1, 2	2false 377	1 ⊢ (𝜑 ↔ ⊥)

Syntax hints:  ¬ wn 3   ↔ wb 207  ⊥wfal 1534

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 208  df-tru 1525  df-fal 1535

This theorem is referenced by:  falantru  1557  trunortru  1571  trunorfal  1572  ralnralall  4372  tgcgr4  25999  frgrregord013  27866  nrmo  33367  bj-df-nul  33946  bicontr  34890  aibnbaif  42684  aifftbifffaibif  42698  atnaiana  42700