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

Theorem eqbrriv 4463
 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 3793 . . 3 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5 df-br 3793 . . 3 (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
63, 4, 53bitr3i 203 . 2 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
71, 2, 6eqrelriiv 4462 1 𝐴 = 𝐵
 Colors of variables: wff set class Syntax hints:   ↔ wb 102   = wceq 1259   ∈ wcel 1409  ⟨cop 3406   class class class wbr 3792  Rel wrel 4378 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380 This theorem is referenced by:  resco  4853  tpostpos  5910
 Copyright terms: Public domain W3C validator