Users' Mathboxes Mathbox for Rodolfo Medina < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqbrrdv2 Structured version   Visualization version   GIF version

Theorem eqbrrdv2 38864
Description: Other version of eqbrrdiv 5804. (Contributed by Rodolfo Medina, 30-Sep-2010.)
Hypothesis
Ref Expression
eqbrrdv2.1 (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦𝑥𝐵𝑦))
Assertion
Ref Expression
eqbrrdv2 (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦

Proof of Theorem eqbrrdv2
StepHypRef Expression
1 eqbrrdv2.1 . . . 4 (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦𝑥𝐵𝑦))
2 df-br 5144 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3 df-br 5144 . . . 4 (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
41, 2, 33bitr3g 313 . . 3 (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
54eqrelrdv2 5805 . 2 (((Rel 𝐴 ∧ Rel 𝐵) ∧ ((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑)) → 𝐴 = 𝐵)
65anabss5 668 1 (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  cop 4632   class class class wbr 5143  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692
This theorem is referenced by:  prter3  38883
  Copyright terms: Public domain W3C validator