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Theorem 3brtr3i 4714
 Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
Hypotheses
Ref Expression
3brtr3.1 𝐴𝑅𝐵
3brtr3.2 𝐴 = 𝐶
3brtr3.3 𝐵 = 𝐷
Assertion
Ref Expression
3brtr3i 𝐶𝑅𝐷

Proof of Theorem 3brtr3i
StepHypRef Expression
1 3brtr3.2 . . 3 𝐴 = 𝐶
2 3brtr3.1 . . 3 𝐴𝑅𝐵
31, 2eqbrtrri 4708 . 2 𝐶𝑅𝐵
4 3brtr3.3 . 2 𝐵 = 𝐷
53, 4breqtri 4710 1 𝐶𝑅𝐷
 Colors of variables: wff setvar class Syntax hints:   = wceq 1523   class class class wbr 4685 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686 This theorem is referenced by:  supsrlem  9970  ef01bndlem  14958  pige3  24314  log2ublem1  24718  log2ub  24721  ppiublem1  24972  logfacrlim2  24996  chebbnd1  25206  nmoptri2i  29086  dpmul4  29750  problem5  31689  fouriersw  40766
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