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Theorem dcned 2255
Description: Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
Hypothesis
Ref Expression
dcned.eq  |-  ( ph  -> DECID  A  =  B )
Assertion
Ref Expression
dcned  |-  ( ph  -> DECID  A  =/=  B )

Proof of Theorem dcned
StepHypRef Expression
1 dcned.eq . . 3  |-  ( ph  -> DECID  A  =  B )
2 dcn 780 . . 3  |-  (DECID  A  =  B  -> DECID  -.  A  =  B )
31, 2syl 14 . 2  |-  ( ph  -> DECID  -.  A  =  B )
4 df-ne 2250 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
54dcbii 781 . 2  |-  (DECID  A  =/= 
B  <-> DECID  -.  A  =  B
)
63, 5sylibr 132 1  |-  ( ph  -> DECID  A  =/=  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  DECID wdc 776    = wceq 1285    =/= wne 2249
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-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777  df-ne 2250
This theorem is referenced by:  nn0n0n1ge2b  8560  algcvgblem  10638
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