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Theorem dfnot 1303
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 1292 . 2 ¬ ⊥
2 mtt 643 . 2 (¬ ⊥ → (¬ 𝜑 ↔ (𝜑 → ⊥)))
31, 2ax-mp 7 1 𝜑 ↔ (𝜑 → ⊥))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wfal 1290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291
This theorem is referenced by:  inegd  1304  pclem6  1306  alnex  1429  alexim  1577  difin  3218  indifdir  3237  recvguniq  10107  bj-axempty2  10977
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