Theorem bifal 1576

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 1574	. 2 ⊢ ¬ ⊥
3	1, 2	2false 377	1 ⊢ (𝜑 ↔ ⊥)

Syntax hints:  ¬ wn 3   ↔ wb 208  ⊥wfal 1572

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 1563  df-fal 1573

This theorem is referenced by:  falantru  1595  dfnul4  4287  dfnul2  4288  abf  4360  ralnralall  4467  tgcgr4  28697  frgrregord013  30594  nrmo  36767  bj-ntrufal  37009  bicontr  38576  aibnbaif  47498  aifftbifffaibif  47512  atnaiana  47514