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Theorem dcn 757
Description: A decidable proposition is decidable when negated. (Contributed by Jim Kingdon, 25-Mar-2018.)
Assertion
Ref Expression
dcn (DECID 𝜑DECID ¬ 𝜑)

Proof of Theorem dcn
StepHypRef Expression
1 notnot 569 . . . 4 (𝜑 → ¬ ¬ 𝜑)
21orim2i 688 . . 3 ((¬ 𝜑𝜑) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
32orcoms 659 . 2 ((𝜑 ∨ ¬ 𝜑) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4 df-dc 754 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5 df-dc 754 . 2 (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
63, 4, 53imtr4i 194 1 (DECID 𝜑DECID ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  pm5.18dc  788  pm4.67dc  792  pm2.54dc  801  imordc  807  pm4.54dc  816  stabtestimpdc  835  annimdc  856  pm4.55dc  857  pm3.12dc  876  pm3.13dc  877  dn1dc  878  xor3dc  1294  dfbi3dc  1304  dcned  2226
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