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Theorem 3brtr3g 4243
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 4221 . 2  |-  ( A R B  <->  C R D )
51, 4sylib 189 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:  syl5eqbrr  4246  syl6breq  4251  ssenen  7281  adderpq  8833  mulerpq  8834  ltaddnq  8851  ege2le3  12692  ovolfiniun  19397  dvfsumlem3  19912  basellem9  20871  pnt2  21307  pnt  21308  siilem1  22352
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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