Theorem bifal 1557

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 1555	. 2 ⊢ ¬ ⊥
3	1, 2	2false 375	1 ⊢ (𝜑 ↔ ⊥)

Syntax hints:  ¬ wn 3   ↔ wb 206  ⊥wfal 1553

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 1544  df-fal 1554

This theorem is referenced by:  falantru  1576  dfnul4  4285  dfnul2  4286  abf  4356  ralnralall  4464  tgcgr4  28552  frgrregord013  30419  nrmo  36553  bj-ntrufal  36712  bicontr  38220  aibnbaif  47095  aifftbifffaibif  47109  atnaiana  47111