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| Description: A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.) | 
| Ref | Expression | 
|---|---|
| bifal.1 | ⊢ ¬ 𝜑 | 
| Ref | Expression | 
|---|---|
| bifal | ⊢ (𝜑 ↔ ⊥) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bifal.1 | . 2 ⊢ ¬ 𝜑 | |
| 2 | fal 1554 | . 2 ⊢ ¬ ⊥ | |
| 3 | 1, 2 | 2false 375 | 1 ⊢ (𝜑 ↔ ⊥) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ⊥wfal 1552 | 
| 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 1543 df-fal 1553 | 
| This theorem is referenced by: falantru 1575 dfnul4 4335 dfnul2 4336 abf 4406 ralnralall 4515 tgcgr4 28539 frgrregord013 30414 nrmo 36411 bj-ntrufal 36570 bicontr 38087 aibnbaif 46919 aifftbifffaibif 46933 atnaiana 46935 | 
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