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Theorem neeq2 2378
Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
Assertion
Ref Expression
neeq2  |-  ( A  =  B  ->  ( C  =/=  A  <->  C  =/=  B ) )

Proof of Theorem neeq2
StepHypRef Expression
1 eqeq2 2203 . . 3  |-  ( A  =  B  ->  ( C  =  A  <->  C  =  B ) )
21notbid 668 . 2  |-  ( A  =  B  ->  ( -.  C  =  A  <->  -.  C  =  B ) )
3 df-ne 2365 . 2  |-  ( C  =/=  A  <->  -.  C  =  A )
4 df-ne 2365 . 2  |-  ( C  =/=  B  <->  -.  C  =  B )
52, 3, 43bitr4g 223 1  |-  ( A  =  B  ->  ( C  =/=  A  <->  C  =/=  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 105    = wceq 1364    =/= wne 2364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-ne 2365
This theorem is referenced by:  neeq2i  2380  neeq2d  2383  disji2  4022  fodjuomnilemdc  7193  netap  7304  2oneel  7306  2omotaplemap  7307  2omotaplemst  7308  exmidapne  7310  xrlttri3  9853  isnzr2  13664  neapmkv  15503  neap0mkv  15504  ltlenmkv  15505
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