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Theorem 3brtr3g 5136
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 5112 . 2 (𝐴𝑅𝐵𝐶𝑅𝐷)
51, 4sylib 217 1 (𝜑𝐶𝑅𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541   class class class wbr 5103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104
This theorem is referenced by:  eqbrtrrid  5139  breqtrdi  5144  ssenen  9091  adderpq  10888  mulerpq  10889  ltaddnq  10906  ege2le3  15964  ovolfiniun  24849  dvfsumlem3  25376  basellem9  26422  pnt2  26945  pnt  26946  siilem1  29679  omndaddr  31798  ogrpaddltrd  31810  sn-0ne2  40813
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