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Theorem 3brtr4g 4055
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 4032 . 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 1623   class class class wbr 4023
This theorem is referenced by:  syl5eqbr  4056  limensuci  7037  infensuc  7039  rlimneg  12120  isumsup2  12305  crt  12846  4sqlem6  12990  gzrngunit  16437  ovolunlem1a  18855  ovolfiniun  18860  ioombl1lem1  18915  ioombl1lem4  18918  iblss  19159  itgle  19164  dvfsumlem3  19375  emcllem6  20294  pntpbnd1a  20734  ostth2lem4  20785  dalem-cly  29860  dalem10  29862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024
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