Theorem eqbrriv 4850

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 4115	. . 3 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5		df-br 4115	. . 3 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
6	3, 4, 5	3bitr3i 210	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
7	1, 2, 6	eqrelriiv 4849	1 ⊢ 𝐴 = 𝐵

Syntax hints:   ↔ wb 105   = wceq 1398   ∈ wcel 2205  ⟨cop 3697   class class class wbr 4114  Rel wrel 4759

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761

This theorem is referenced by:  resco  5272  tpostpos  6508