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Theorem 3brtr3g 4656
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 4632 . 2 (𝐴𝑅𝐵𝐶𝑅𝐷)
51, 4sylib 208 1 (𝜑𝐶𝑅𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480   class class class wbr 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624
This theorem is referenced by:  syl5eqbrr  4659  syl6breq  4664  ssenen  8094  adderpq  9738  mulerpq  9739  ltaddnq  9756  ege2le3  14764  ovolfiniun  23209  dvfsumlem3  23729  basellem9  24749  pnt2  25236  pnt  25237  siilem1  27594  omndaddr  29534  ogrpaddltrd  29547
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