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Theorem 3brtr4g 3843
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
Hypotheses
Ref Expression
3brtr4g.1  |-  ( ph  ->  A R B )
3brtr4g.2  |-  C  =  A
3brtr4g.3  |-  D  =  B
Assertion
Ref Expression
3brtr4g  |-  ( ph  ->  C R D )

Proof of Theorem 3brtr4g
StepHypRef Expression
1 3brtr4g.1 . 2  |-  ( ph  ->  A R B )
2 3brtr4g.2 . . 3  |-  C  =  A
3 3brtr4g.3 . . 3  |-  D  =  B
42, 3breq12i 3820 . 2  |-  ( C R D  <->  A R B )
51, 4sylibr 132 1  |-  ( ph  ->  C R D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   class class class wbr 3811
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812
This theorem is referenced by:  syl5eqbr  3844  crth  10980
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