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Theorem dcne 2257
Description: Decidable equality expressed in terms of  =/=. Basically the same as df-dc 777. (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 777 . 2  |-  (DECID  A  =  B  <->  ( A  =  B  \/  -.  A  =  B ) )
2 df-ne 2247 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
32orbi2i 712 . 2  |-  ( ( A  =  B  \/  A  =/=  B )  <->  ( A  =  B  \/  -.  A  =  B )
)
41, 3bitr4i 185 1  |-  (DECID  A  =  B  <->  ( A  =  B  \/  A  =/= 
B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 103    \/ wo 662  DECID wdc 776    = wceq 1285    =/= wne 2246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777  df-ne 2247
This theorem is referenced by:  zdceq  8504  nn0lt2  8510  qdceq  9333  nn0seqcvgd  10567
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