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Theorem bj-opelopabid 37189
Description: Membership in an ordered-pair class abstraction. One can remove the DV condition on 𝑥, 𝑦 by using opabid 5529 in place of opabidw 5528. (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 5143 . 2 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
2 opabidw 5528 . 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 2107  cop 4631   class class class wbr 5142  {copab 5204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205
This theorem is referenced by:  bj-opabco  37190
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