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Theorem bj-opelopabid 37170
Description: Membership in an ordered-pair class abstraction. One can remove the DV condition on 𝑥, 𝑦 by using opabid 5535 in place of opabidw 5534. (Contributed by BJ, 22-May-2024.)
Assertion
Ref Expression
bj-opelopabid (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-opelopabid
StepHypRef Expression
1 df-br 5149 . 2 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2 opabidw 5534 . 2 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
31, 2bitri 275 1 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2106  cop 4637   class class class wbr 5148  {copab 5210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211
This theorem is referenced by:  bj-opabco  37171
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