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Theorem 3brtr3g 3961
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
Hypotheses
Ref Expression
3brtr3g.1  |-  ( ph  ->  A R B )
3brtr3g.2  |-  A  =  C
3brtr3g.3  |-  B  =  D
Assertion
Ref Expression
3brtr3g  |-  ( ph  ->  C R D )

Proof of Theorem 3brtr3g
StepHypRef Expression
1 3brtr3g.1 . 2  |-  ( ph  ->  A R B )
2 3brtr3g.2 . . 3  |-  A  =  C
3 3brtr3g.3 . . 3  |-  B  =  D
42, 3breq12i 3938 . 2  |-  ( A R B  <->  C R D )
51, 4sylib 121 1  |-  ( ph  ->  C R D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   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-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-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930
This theorem is referenced by:  eqbrtrrid  3964  breqtrdi  3969  ssenen  6745  endjusym  6981  djuen  7067  ege2le3  11384
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