Theorem bifal 1327

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 1322	. 2 ⊢ ¬ ⊥
3	1, 2	2false 339	1 ⊢ (φ ↔ ⊥ )

Syntax hints:  ¬ wn 3   ↔ wb 176   ⊥ wfal 1317

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 177  df-tru 1319  df-fal 1320

This theorem is referenced by:  truanfal  1337  falantru  1338  trubifal  1351  spfalwOLD  1699