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Theorem dfnot 1261
Description: Given falsum, we can define the negation of a wff φ as the statement that a contradiction follows from assuming φ. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
Assertion
Ref Expression
dfnot φ ↔ (φ → ⊥ ))

Proof of Theorem dfnot
StepHypRef Expression
1 fal 1249 . 2 ¬ ⊥
2 mtt 609 . 2 (¬ ⊥ → (¬ φ ↔ (φ → ⊥ )))
31, 2ax-mp 7 1 φ ↔ (φ → ⊥ ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98  wfal 1247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248
This theorem is referenced by:  inegd  1262  pclem6  1264  alnex  1385  alexim  1533  difin  3168  indifdir  3187
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