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Theorem brabidgaw 35770
Description: The law of concretion for a binary relation. Special case of brabga 5389. Version of brabidga 35771 with a disjoint variable condition, which does not require ax-13 2382. (Contributed by Peter Mazsa, 24-Nov-2018.) (Revised by Gino Giotto, 2-Apr-2024.)
Hypothesis
Ref Expression
brabidgaw.1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Assertion
Ref Expression
brabidgaw (𝑥𝑅𝑦𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem brabidgaw
StepHypRef Expression
1 brabidgaw.1 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
21breqi 5039 . 2 (𝑥𝑅𝑦𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦)
3 df-br 5034 . 2 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4 opabidw 5380 . 2 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
52, 3, 43bitri 300 1 (𝑥𝑅𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2112  cop 4534   class class class wbr 5033  {copab 5095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096
This theorem is referenced by:  inxpxrn  35796
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