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Theorem dcne 2211
Description: Decidable equality expressed in terms of . Basically the same as df-dc 742. (Contributed by Jim Kingdon, 14-Mar-2020.)
Assertion
Ref Expression
dcne (DECID A = B ↔ (A = B AB))

Proof of Theorem dcne
StepHypRef Expression
1 df-dc 742 . 2 (DECID A = B ↔ (A = B ¬ A = B))
2 df-ne 2203 . . 3 (AB ↔ ¬ A = B)
32orbi2i 678 . 2 ((A = B AB) ↔ (A = B ¬ A = B))
41, 3bitr4i 176 1 (DECID A = B ↔ (A = B AB))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98   wo 628  DECID wdc 741   = wceq 1242  wne 2201
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 629
This theorem depends on definitions:  df-bi 110  df-dc 742  df-ne 2203
This theorem is referenced by:  zdceq  8072  nn0lt2  8078
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