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Theorem eqnbrtrd 4005
Description: Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypotheses
Ref Expression
eqnbrtrd.1  |-  ( ph  ->  A  =  B )
eqnbrtrd.2  |-  ( ph  ->  -.  B R C )
Assertion
Ref Expression
eqnbrtrd  |-  ( ph  ->  -.  A R C )

Proof of Theorem eqnbrtrd
StepHypRef Expression
1 eqnbrtrd.2 . 2  |-  ( ph  ->  -.  B R C )
2 eqnbrtrd.1 . . 3  |-  ( ph  ->  A  =  B )
32breq1d 3997 . 2  |-  ( ph  ->  ( A R C  <-> 
B R C ) )
41, 3mtbird 668 1  |-  ( ph  ->  -.  A R C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1348   class class class wbr 3987
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988
This theorem is referenced by:  xnn0dcle  9746  pczndvds  12256  pcadd  12280
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