Theorem bifal 1559

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 1557	. 2 ⊢ ¬ ⊥
3	1, 2	2false 379	1 ⊢ (𝜑 ↔ ⊥)

Syntax hints:  ¬ wn 3   ↔ wb 209  ⊥wfal 1555

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 210  df-tru 1546  df-fal 1556

This theorem is referenced by:  falantru  1578  trunortruOLD  1593  trunorfalOLD  1595  dfnul4  4239  dfnul2  4240  dfnul4OLD  4244  abf  4317  ralnralall  4430  tgcgr4  26622  frgrregord013  28478  nrmo  34336  bj-ntrufal  34487  bicontr  35975  aibnbaif  44074  aifftbifffaibif  44088  atnaiana  44090