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Theorem dcne 2214
Description: Decidable equality expressed in terms of . Basically the same as df-dc 743. (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 743 . 2 (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
2 df-ne 2206 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
32orbi2i 679 . 2 ((𝐴 = 𝐵𝐴𝐵) ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵))
41, 3bitr4i 176 1 (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98  wo 629  DECID wdc 742   = wceq 1243  wne 2204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630
This theorem depends on definitions:  df-bi 110  df-dc 743  df-ne 2206
This theorem is referenced by:  zdceq  8264  nn0lt2  8270  nn0seqcvgd  9733
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