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Theorem nbrne2 3863
Description: Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
Assertion
Ref Expression
nbrne2  |-  ( ( A R C  /\  -.  B R C )  ->  A  =/=  B
)

Proof of Theorem nbrne2
StepHypRef Expression
1 breq1 3848 . . . 4  |-  ( A  =  B  ->  ( A R C  <->  B R C ) )
21biimpcd 157 . . 3  |-  ( A R C  ->  ( A  =  B  ->  B R C ) )
32necon3bd 2298 . 2  |-  ( A R C  ->  ( -.  B R C  ->  A  =/=  B ) )
43imp 122 1  |-  ( ( A R C  /\  -.  B R C )  ->  A  =/=  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1289    =/= wne 2255   class class class wbr 3845
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846
This theorem is referenced by: (None)
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