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Theorem brintclab 13786
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 4686 . 2 (𝐴 {𝑥𝜑}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {𝑥𝜑})
2 opex 4962 . . 3 𝐴, 𝐵⟩ ∈ V
32elintab 4519 . 2 (⟨𝐴, 𝐵⟩ ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑥))
41, 3bitri 264 1 (𝐴 {𝑥𝜑}𝐵 ↔ ∀𝑥(𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wcel 2030  {cab 2637  cop 4216   cint 4507   class class class wbr 4685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-int 4508  df-br 4686
This theorem is referenced by:  brtrclfv  13787
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