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Theorem eqbrriv 4602
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 3898 . . 3 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5 df-br 3898 . . 3 (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
63, 4, 53bitr3i 209 . 2 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
71, 2, 6eqrelriiv 4601 1 𝐴 = 𝐵
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1314  wcel 1463  cop 3498   class class class wbr 3897  Rel wrel 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514
This theorem is referenced by:  resco  5011  tpostpos  6127
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