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Theorem brintclab 13692
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 4624 . 2 (𝐴 {𝑥𝜑}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {𝑥𝜑})
2 opex 4903 . . 3 𝐴, 𝐵⟩ ∈ V
32elintab 4459 . 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 1478  wcel 1987  {cab 2607  cop 4161   cint 4447   class class class wbr 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-int 4448  df-br 4624
This theorem is referenced by:  brtrclfv  13693
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