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Theorem dcne 2296
Description: Decidable equality expressed in terms of . Basically the same as df-dc 805. (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 805 . 2 (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
2 df-ne 2286 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
32orbi2i 736 . 2 ((𝐴 = 𝐵𝐴𝐵) ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
41, 3bitr4i 186 1 (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 104  wo 682  DECID wdc 804   = wceq 1316  wne 2285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683
This theorem depends on definitions:  df-bi 116  df-dc 805  df-ne 2286
This theorem is referenced by:  updjudhf  6932  zdceq  9094  nn0lt2  9100  xlesubadd  9634  qdceq  9992  xrmaxadd  10998  nn0seqcvgd  11649
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