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Theorem opabresex2d 7486
Description: Obsolete version of opabresex2 7485 as of 13-Dec-2024. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
opabresex2d.1 ((𝜑𝑥(𝑊𝐺)𝑦) → 𝜓)
opabresex2d.2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ 𝑉)
Assertion
Ref Expression
opabresex2d (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ∈ V)
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜃(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opabresex2d
StepHypRef Expression
1 opabresex2d.1 . . . 4 ((𝜑𝑥(𝑊𝐺)𝑦) → 𝜓)
21ex 412 . . 3 (𝜑 → (𝑥(𝑊𝐺)𝑦𝜓))
32alrimivv 1926 . 2 (𝜑 → ∀𝑥𝑦(𝑥(𝑊𝐺)𝑦𝜓))
4 opabresex2d.2 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ 𝑉)
5 opabbrex 7484 . 2 ((∀𝑥𝑦(𝑥(𝑊𝐺)𝑦𝜓) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ∈ V)
63, 4, 5syl2anc 584 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535  wcel 2106  Vcvv 3478   class class class wbr 5148  {copab 5210  cfv 6563
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-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-opab 5211
This theorem is referenced by:  mptmpoopabbrdOLDOLD  8107
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