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Theorem dcne 2260
 Description: Decidable equality expressed in terms of ≠. Basically the same as df-dc 777. (Contributed by Jim Kingdon, 14-Mar-2020.)
Assertion
Ref Expression
dcne (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵𝐴𝐵))

Proof of Theorem dcne
StepHypRef Expression
1 df-dc 777 . 2 (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
2 df-ne 2250 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
32orbi2i 712 . 2 ((𝐴 = 𝐵𝐴𝐵) ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
41, 3bitr4i 185 1 (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵𝐴𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 103   ∨ wo 662  DECID wdc 776   = wceq 1285   ≠ wne 2249 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-io 663 This theorem depends on definitions:  df-bi 115  df-dc 777  df-ne 2250 This theorem is referenced by:  updjudhf  6583  zdceq  8590  nn0lt2  8596  qdceq  9419  nn0seqcvgd  10664
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