Theorem bifal 1586

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 1584	. 2 ⊢ ¬ ⊥
3	1, 2	2false 378	1 ⊢ (𝜑 ↔ ⊥)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ⊥wfal 1582

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-tru 1573  df-fal 1583

This theorem is used by:  falantru  1605  dfnul4  4288  dfnul2  4289  abf  4371  ralnralall  4476  tgcgr4  28851  frgrregord013  30817  nrmo  36978  bj-ntrufal  37219  bicontr  38789  aibnbaif  47702  aifftbifffaibif  47716  atnaiana  47718