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

Theorem eqbrriv 4378
Description: Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
Hypotheses
Ref Expression
eqbrriv.1 Rel A
eqbrriv.2 Rel B
eqbrriv.3 (xAyxBy)
Assertion
Ref Expression
eqbrriv A = B
Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem eqbrriv
StepHypRef Expression
1 eqbrriv.1 . 2 Rel A
2 eqbrriv.2 . 2 Rel B
3 eqbrriv.3 . . 3 (xAyxBy)
4 df-br 3756 . . 3 (xAy ↔ ⟨x, y A)
5 df-br 3756 . . 3 (xBy ↔ ⟨x, y B)
63, 4, 53bitr3i 199 . 2 (⟨x, y A ↔ ⟨x, y B)
71, 2, 6eqrelriiv 4377 1 A = B
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242   wcel 1390  cop 3370   class class class wbr 3755  Rel wrel 4293
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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
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-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295
This theorem is referenced by:  resco  4768  tpostpos  5820
  Copyright terms: Public domain W3C validator