Theorem bifal 1556

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 1554	. 2 ⊢ ¬ ⊥
3	1, 2	2false 375	1 ⊢ (𝜑 ↔ ⊥)

Syntax hints:  ¬ wn 3   ↔ wb 206  ⊥wfal 1552

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 207  df-tru 1543  df-fal 1553

This theorem is referenced by:  falantru  1575  dfnul4  4335  dfnul2  4336  abf  4406  ralnralall  4515  tgcgr4  28539  frgrregord013  30414  nrmo  36411  bj-ntrufal  36570  bicontr  38087  aibnbaif  46919  aifftbifffaibif  46933  atnaiana  46935