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Theorem brabidga 35498
Description: The law of concretion for a binary relation. Special case of brabga 5412. (Contributed by Peter Mazsa, 24-Nov-2018.)
Hypothesis
Ref Expression
brabidga.1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Assertion
Ref Expression
brabidga (𝑥𝑅𝑦𝜑)

Proof of Theorem brabidga
StepHypRef Expression
1 brabidga.1 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
21breqi 5063 . 2 (𝑥𝑅𝑦𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦)
3 df-br 5058 . 2 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4 opabid 5404 . 2 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
52, 3, 43bitri 298 1 (𝑥𝑅𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  cop 4563   class class class wbr 5057  {copab 5119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120
This theorem is referenced by:  inxpxrn  35523
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