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Theorem 3brtr4g 4071
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 4048 . 2  |-  ( C R D  <->  A R B )
51, 4sylibr 203 1  |-  ( ph  ->  C R D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   class class class wbr 4039
This theorem is referenced by:  syl5eqbr  4072  limensuci  7053  infensuc  7055  rlimneg  12136  isumsup2  12321  crt  12862  4sqlem6  13006  gzrngunit  16453  ovolunlem1a  18871  ovolfiniun  18876  ioombl1lem1  18931  ioombl1lem4  18934  iblss  19175  itgle  19180  dvfsumlem3  19391  emcllem6  20310  pntpbnd1a  20750  ostth2lem4  20801  itg2gt0cn  25006  dalem-cly  30482  dalem10  30484
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040
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