Theorem bifal 1497

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 1490	. 2 ⊢ ¬ ⊥
3	1, 2	2false 365	1 ⊢ (𝜑 ↔ ⊥)

Syntax hints:  ¬ wn 3   ↔ wb 196  ⊥wfal 1488

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 197  df-tru 1486  df-fal 1489

This theorem is referenced by:  falantru  1508  ralnralall  4080  tgcgr4  25426  frgrregord013  27253  bj-df-nul  33017  bicontr  33879  aibnbaif  41074  aifftbifffaibif  41088  atnaiana  41090