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

Theorem 3brtr3i 4421
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 4415 . 2  |-  C R B
4 3brtr3.3 . 2  |-  B  =  D
53, 4breqtri 4417 1  |-  C R D
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405   class class class wbr 4394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395
This theorem is referenced by:  supsrlem  9438  ef01bndlem  14020  pige3  23094  log2ublem1  23494  log2ub  23497  ppiublem1  23750  logfacrlim2  23774  chebbnd1  23930  nmoptri2i  27311  problem5  29756  fouriersw  37364
  Copyright terms: Public domain W3C validator