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Theorem 3brtr4g 4244
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 4221 . 2  |-  ( C R D  <->  A R B )
51, 4sylibr 204 1  |-  ( ph  ->  C R D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   class class class wbr 4212
This theorem is referenced by:  syl5eqbr  4245  limensuci  7283  infensuc  7285  rlimneg  12440  isumsup2  12626  crt  13167  4sqlem6  13311  gzrngunit  16764  ovolunlem1a  19392  ovolfiniun  19397  ioombl1lem1  19452  ioombl1lem4  19455  iblss  19696  itgle  19701  dvfsumlem3  19912  emcllem6  20839  pntpbnd1a  21279  ostth2lem4  21330  itg2gt0cn  26260  dalem-cly  30468  dalem10  30470
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213
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