Theorem bifal 1555

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 1553	. 2 ⊢ ¬ ⊥
3	1, 2	2false 376	1 ⊢ (𝜑 ↔ ⊥)

Syntax hints:  ¬ wn 3   ↔ wb 205  ⊥wfal 1551

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 206  df-tru 1542  df-fal 1552

This theorem is referenced by:  falantru  1574  trunorfalOLD  1590  dfnul4  4258  dfnul2  4259  dfnul4OLD  4263  abf  4336  ralnralall  4449  tgcgr4  26892  frgrregord013  28759  nrmo  34599  bj-ntrufal  34750  bicontr  36238  aibnbaif  44402  aifftbifffaibif  44416  atnaiana  44418