Theorem eqbrriv 4775

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 4049	. . 3 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5		df-br 4049	. . 3 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
6	3, 4, 5	3bitr3i 210	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
7	1, 2, 6	eqrelriiv 4774	1 ⊢ 𝐴 = 𝐵

Syntax hints:   ↔ wb 105   = wceq 1373   ∈ wcel 2177  ⟨cop 3638   class class class wbr 4048  Rel wrel 4685

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-br 4049  df-opab 4111  df-xp 4686  df-rel 4687

This theorem is referenced by:  resco  5193  tpostpos  6360