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Theorem opelopabb 34954
Description: Membership of an ordered pair in a class abstraction of ordered pairs, biconditional form. (Contributed by BJ, 17-Dec-2023.)
Hypotheses
Ref Expression
opelopabb.xph (𝜑 → ∀𝑥𝜑)
opelopabb.yph (𝜑 → ∀𝑦𝜑)
opelopabb.xch (𝜑 → Ⅎ𝑥𝜒)
opelopabb.ych (𝜑 → Ⅎ𝑦𝜒)
opelopabb.is ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
opelopabb (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem opelopabb
StepHypRef Expression
1 elopab 5382 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
2 opelopabb.xph . . 3 (𝜑 → ∀𝑥𝜑)
3 opelopabb.yph . . 3 (𝜑 → ∀𝑦𝜑)
4 opelopabb.xch . . 3 (𝜑 → Ⅎ𝑥𝜒)
5 opelopabb.ych . . 3 (𝜑 → Ⅎ𝑦𝜒)
6 opelopabb.is . . 3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
72, 3, 4, 5, 6copsex2b 34952 . 2 (𝜑 → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
81, 7syl5bb 286 1 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1540   = wceq 1542  wex 1786  wnf 1790  wcel 2114  Vcvv 3398  cop 4522  {copab 5092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-dif 3846  df-un 3848  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-opab 5093
This theorem is referenced by:  opelopabbv  34955
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