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Theorem dcned 2312
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 827 . . 3  |-  (DECID  A  =  B  -> DECID  -.  A  =  B )
31, 2syl 14 . 2  |-  ( ph  -> DECID  -.  A  =  B )
4 df-ne 2307 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
54dcbii 825 . 2  |-  (DECID  A  =/= 
B  <-> DECID  -.  A  =  B
)
63, 5sylibr 133 1  |-  ( ph  -> DECID  A  =/=  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  DECID wdc 819    = wceq 1331    =/= wne 2306
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-in1 603  ax-in2 604  ax-io 698
This theorem depends on definitions:  df-bi 116  df-dc 820  df-ne 2307
This theorem is referenced by:  nn0n0n1ge2b  9123  algcvgblem  11719
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