Theorem bifal 1563

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 1561	. 2 ⊢ ¬ ⊥
3	1, 2	2false 376	1 ⊢ (𝜑 ↔ ⊥)

Syntax hints:  ¬ wn 3   ↔ wb 207  ⊥wfal 1559

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 1550  df-fal 1560

This theorem is referenced by:  falantru  1582  dfnul4  4263  dfnul2  4264  abf  4334  ralnralall  4441  tgcgr4  28617  frgrregord013  30483  nrmo  36638  bj-ntrufal  36880  bicontr  38447  aibnbaif  47370  aifftbifffaibif  47384  atnaiana  47386