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Theorem 3brtr3g 5186
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
Hypotheses
Ref Expression
3brtr3g.1 (𝜑𝐴𝑅𝐵)
3brtr3g.2 𝐴 = 𝐶
3brtr3g.3 𝐵 = 𝐷
Assertion
Ref Expression
3brtr3g (𝜑𝐶𝑅𝐷)

Proof of Theorem 3brtr3g
StepHypRef Expression
1 3brtr3g.1 . 2 (𝜑𝐴𝑅𝐵)
2 3brtr3g.2 . . 3 𝐴 = 𝐶
3 3brtr3g.3 . . 3 𝐵 = 𝐷
42, 3breq12i 5162 . 2 (𝐴𝑅𝐵𝐶𝑅𝐷)
51, 4sylib 217 1 (𝜑𝐶𝑅𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534   class class class wbr 5153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154
This theorem is referenced by:  eqbrtrrid  5189  breqtrdi  5194  ssenen  9189  adderpq  10999  mulerpq  11000  ltaddnq  11017  ege2le3  16092  ovolfiniun  25521  dvfsumlem3  26054  basellem9  27117  pnt2  27642  pnt  27643  siilem1  30784  omndaddr  32942  ogrpaddltrd  32954  sn-0ne2  42186
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