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Theorem 3brtr4g 4617
 Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
Hypotheses
Ref Expression
3brtr4g.1 (𝜑𝐴𝑅𝐵)
3brtr4g.2 𝐶 = 𝐴
3brtr4g.3 𝐷 = 𝐵
Assertion
Ref Expression
3brtr4g (𝜑𝐶𝑅𝐷)

Proof of Theorem 3brtr4g
StepHypRef Expression
1 3brtr4g.1 . 2 (𝜑𝐴𝑅𝐵)
2 3brtr4g.2 . . 3 𝐶 = 𝐴
3 3brtr4g.3 . . 3 𝐷 = 𝐵
42, 3breq12i 4592 . 2 (𝐶𝑅𝐷𝐴𝑅𝐵)
51, 4sylibr 223 1 (𝜑𝐶𝑅𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   class class class wbr 4583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584 This theorem is referenced by:  syl5eqbr  4618  limensuci  8021  infensuc  8023  rlimneg  14225  isumsup2  14417  crth  15321  4sqlem6  15485  gzrngunit  19631  matgsum  20062  ovolunlem1a  23071  ovolfiniun  23076  ioombl1lem1  23133  ioombl1lem4  23136  iblss  23377  itgle  23382  dvfsumlem3  23595  emcllem6  24527  gausslemma2dlem0f  24886  gausslemma2dlem0g  24887  pntpbnd1a  25074  ostth2lem4  25125  omsmon  29687  itg2gt0cn  32635  dalem-cly  33975  dalem10  33977  fourierdlem103  39102  fourierdlem104  39103
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