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Theorem 3brtr3g 4819
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 4795 . 2 (𝐴𝑅𝐵𝐶𝑅𝐷)
51, 4sylib 208 1 (𝜑𝐶𝑅𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631   class class class wbr 4786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787
This theorem is referenced by:  syl5eqbrr  4822  syl6breq  4827  ssenen  8290  adderpq  9980  mulerpq  9981  ltaddnq  9998  ege2le3  15026  ovolfiniun  23489  dvfsumlem3  24011  basellem9  25036  pnt2  25523  pnt  25524  siilem1  28046  omndaddr  30047  ogrpaddltrd  30060
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