Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqbrrdv GIF version

Theorem eqbrrdv 4680
 Description: Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
eqbrrdv.1 (𝜑 → Rel 𝐴)
eqbrrdv.2 (𝜑 → Rel 𝐵)
eqbrrdv.3 (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))
Assertion
Ref Expression
eqbrrdv (𝜑𝐴 = 𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦

Proof of Theorem eqbrrdv
StepHypRef Expression
1 eqbrrdv.3 . . . 4 (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))
2 df-br 3966 . . . 4 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3 df-br 3966 . . . 4 (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
41, 2, 33bitr3g 221 . . 3 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
54alrimivv 1855 . 2 (𝜑 → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
6 eqbrrdv.1 . . 3 (𝜑 → Rel 𝐴)
7 eqbrrdv.2 . . 3 (𝜑 → Rel 𝐵)
8 eqrel 4672 . . 3 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
96, 7, 8syl2anc 409 . 2 (𝜑 → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
105, 9mpbird 166 1 (𝜑𝐴 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1333   = wceq 1335   ∈ wcel 2128  ⟨cop 3563   class class class wbr 3965  Rel wrel 4588 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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4589  df-rel 4590 This theorem is referenced by:  eqbrrdva  4753
 Copyright terms: Public domain W3C validator