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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  =  B  \/  A  =/= 
B ) )

Proof of Theorem dcne
StepHypRef Expression
1 df-dc 820 . 2  |-  (DECID  A  =  B  <->  ( A  =  B  \/  -.  A  =  B ) )
2 df-ne 2309 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
32orbi2i 751 . 2  |-  ( ( A  =  B  \/  A  =/=  B )  <->  ( A  =  B  \/  -.  A  =  B )
)
41, 3bitr4i 186 1  |-  (DECID  A  =  B  <->  ( A  =  B  \/  A  =/= 
B ) )
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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