Theorem bifal 1553

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 1551	. 2 ⊢ ¬ ⊥
3	1, 2	2false 378	1 ⊢ (𝜑 ↔ ⊥)

Syntax hints:  ¬ wn 3   ↔ wb 208  ⊥wfal 1549

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 209  df-tru 1540  df-fal 1550

This theorem is referenced by:  falantru  1572  trunortruOLD  1587  trunorfalOLD  1589  ralnralall  4458  tgcgr4  26317  frgrregord013  28174  nrmo  33758  bj-df-nul  34351  bicontr  35373  aibnbaif  43163  aifftbifffaibif  43177  atnaiana  43179