Theorem eqbrriv 4739

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	⊢ (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)

Assertion

Ref	Expression
eqbrriv	⊢ 𝐴 = 𝐵

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem eqbrriv

Step	Hyp	Ref	Expression
1		eqbrriv.1	. 2 ⊢ Rel 𝐴
2		eqbrriv.2	. 2 ⊢ Rel 𝐵
3		eqbrriv.3	. . 3 ⊢ (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)
4		df-br 4019	. . 3 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5		df-br 4019	. . 3 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
6	3, 4, 5	3bitr3i 210	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
7	1, 2, 6	eqrelriiv 4738	1 ⊢ 𝐴 = 𝐵

Syntax hints:   ↔ wb 105   = wceq 1364   ∈ wcel 2160  ⟨cop 3610   class class class wbr 4018  Rel wrel 4649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651

This theorem is referenced by:  resco  5151  tpostpos  6290