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Theorem brintclab 14846
Description: Two ways to express a binary relation which is the intersection of a class. (Contributed by RP, 4-Apr-2020.)
Assertion
Ref Expression
brintclab (𝐴 {𝑥𝜑}𝐵 ↔ ∀𝑥(𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem brintclab
StepHypRef Expression
1 df-br 5105 . 2 (𝐴 {𝑥𝜑}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {𝑥𝜑})
2 opex 5420 . . 3 𝐴, 𝐵⟩ ∈ V
32elintab 4918 . 2 (⟨𝐴, 𝐵⟩ ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑥))
41, 3bitri 275 1 (𝐴 {𝑥𝜑}𝐵 ↔ ∀𝑥(𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540  wcel 2107  {cab 2715  cop 4591   cint 4906   class class class wbr 5104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-int 4907  df-br 5105
This theorem is referenced by:  brtrclfv  14847
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