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Theorem dcne 2273
 Description: Decidable equality expressed in terms of ≠. Basically the same as df-dc 784. (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 784 . 2 (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
2 df-ne 2263 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
32orbi2i 717 . 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 667  DECID wdc 783   = wceq 1296   ≠ wne 2262 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668 This theorem depends on definitions:  df-bi 116  df-dc 784  df-ne 2263 This theorem is referenced by:  updjudhf  6850  zdceq  8920  nn0lt2  8926  xlesubadd  9449  qdceq  9807  xrmaxadd  10804  nn0seqcvgd  11450
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