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Theorem brintclab 14895
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 5110 . 2 (𝐴 {𝑥𝜑}𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {𝑥𝜑})
2 opex 5425 . . 3 𝐴, 𝐵⟩ ∈ V
32elintab 4923 . 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 2710  cop 4596   cint 4911   class class class wbr 5109
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 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
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 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-int 4912  df-br 5110
This theorem is referenced by:  brtrclfv  14896
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