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Theorem 3brtr3g 4054
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 4032 . 2  |-  ( A R B  <->  C R D )
51, 4sylib 188 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:  syl5eqbrr  4057  syl6breq  4062  ssenen  7035  adderpq  8580  mulerpq  8581  ltaddnq  8598  ege2le3  12371  ovolfiniun  18860  dvfsumlem3  19375  basellem9  20326  pnt2  20762  pnt  20763  siilem1  21429
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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