Theorem bifal 1552

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 1550	. 2 ⊢ ¬ ⊥
3	1, 2	2false 378	1 ⊢ (𝜑 ↔ ⊥)

Syntax hints:  ¬ wn 3   ↔ wb 208  ⊥wfal 1548

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 1539  df-fal 1549

This theorem is referenced by:  falantru  1571  trunortruOLD  1586  trunorfalOLD  1588  ralnralall  4461  tgcgr4  26320  frgrregord013  28177  nrmo  33762  bj-df-nul  34352  bicontr  35362  aibnbaif  43150  aifftbifffaibif  43164  atnaiana  43166