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Theorem nbrne2 3948
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 3932 . . . 4  |-  ( A  =  B  ->  ( A R C  <->  B R C ) )
21biimpcd 158 . . 3  |-  ( A R C  ->  ( A  =  B  ->  B R C ) )
32necon3bd 2351 . 2  |-  ( A R C  ->  ( -.  B R C  ->  A  =/=  B ) )
43imp 123 1  |-  ( ( A R C  /\  -.  B R C )  ->  A  =/=  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1331    =/= wne 2308   class class class wbr 3929
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  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930
This theorem is referenced by: (None)
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