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Theorem 3brtr3g 4919
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 4895 . 2 (𝐴𝑅𝐵𝐶𝑅𝐷)
51, 4sylib 210 1 (𝜑𝐶𝑅𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601   class class class wbr 4886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887
This theorem is referenced by:  syl5eqbrr  4922  syl6breq  4927  ssenen  8422  adderpq  10113  mulerpq  10114  ltaddnq  10131  ege2le3  15222  ovolfiniun  23705  dvfsumlem3  24228  basellem9  25267  pnt2  25754  pnt  25755  siilem1  28278  omndaddr  30269  ogrpaddltrd  30282
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