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Theorem 3brtr3g 4434
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 4412 . 2  |-  ( A R B  <->  C R D )
51, 4sylib 196 1  |-  ( ph  ->  C R D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   class class class wbr 4403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404
This theorem is referenced by:  syl5eqbrr  4437  syl6breq  4442  ssenen  7598  adderpq  9239  mulerpq  9240  ltaddnq  9257  ege2le3  13496  ovolfiniun  21119  dvfsumlem3  21636  basellem9  22562  pnt2  22998  pnt  22999  siilem1  24423  omndaddr  26335  ogrpaddltrd  26348  archiabllem2a  26376
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