ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  3brtr4i Structured version   GIF version

Theorem 3brtr4i 3782
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
Hypotheses
Ref Expression
3brtr4.1 A𝑅B
3brtr4.2 𝐶 = A
3brtr4.3 𝐷 = B
Assertion
Ref Expression
3brtr4i 𝐶𝑅𝐷

Proof of Theorem 3brtr4i
StepHypRef Expression
1 3brtr4.2 . . 3 𝐶 = A
2 3brtr4.1 . . 3 A𝑅B
31, 2eqbrtri 3773 . 2 𝐶𝑅B
4 3brtr4.3 . 2 𝐷 = B
53, 4breqtrri 3779 1 𝐶𝑅𝐷
Colors of variables: wff set class
Syntax hints:   = wceq 1242   class class class wbr 3754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755
This theorem is referenced by:  1lt2nq  6382  0lt1sr  6645  ax0lt1  6712  ax-0lt1  6741  declt  8113  decltc  8114  frecfzennn  8822
  Copyright terms: Public domain W3C validator