MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3brtr3i Structured version   Visualization version   Unicode version

Theorem 3brtr3i 4430
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
Hypotheses
Ref Expression
3brtr3.1  |-  A R B
3brtr3.2  |-  A  =  C
3brtr3.3  |-  B  =  D
Assertion
Ref Expression
3brtr3i  |-  C R D

Proof of Theorem 3brtr3i
StepHypRef Expression
1 3brtr3.2 . . 3  |-  A  =  C
2 3brtr3.1 . . 3  |-  A R B
31, 2eqbrtrri 4424 . 2  |-  C R B
4 3brtr3.3 . 2  |-  B  =  D
53, 4breqtri 4426 1  |-  C R D
Colors of variables: wff setvar class
Syntax hints:    = wceq 1444   class class class wbr 4402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403
This theorem is referenced by:  supsrlem  9535  ef01bndlem  14238  pige3  23472  log2ublem1  23872  log2ub  23875  ppiublem1  24130  logfacrlim2  24154  chebbnd1  24310  nmoptri2i  27752  problem5  30301  fouriersw  38095
  Copyright terms: Public domain W3C validator