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Theorem dcne 2319
 Description: Decidable equality expressed in terms of ≠. Basically the same as df-dc 820. (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 820 . 2 (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
2 df-ne 2309 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
32orbi2i 751 . 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 697  DECID wdc 819   = wceq 1331   ≠ wne 2308 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 698 This theorem depends on definitions:  df-bi 116  df-dc 820  df-ne 2309 This theorem is referenced by:  updjudhf  6964  zdceq  9133  nn0lt2  9139  xlesubadd  9673  qdceq  10031  xrmaxadd  11037  nn0seqcvgd  11729
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