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Theorem opelopabbv 34422
 Description: Membership of an ordered pair in a class abstraction of ordered pairs, biconditional form. (Contributed by BJ, 17-Dec-2023.)
Hypotheses
Ref Expression
opelopabbv.def (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
opelopabbv.is ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
opelopabbv (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem opelopabbv
StepHypRef Expression
1 opelopabbv.def . . 3 (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
21eleq2d 2896 . 2 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
3 ax-5 1904 . . 3 (𝜑 → ∀𝑥𝜑)
4 ax-5 1904 . . 3 (𝜑 → ∀𝑦𝜑)
5 nfvd 1909 . . 3 (𝜑 → Ⅎ𝑥𝜒)
6 nfvd 1909 . . 3 (𝜑 → Ⅎ𝑦𝜒)
7 opelopabbv.is . . 3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
83, 4, 5, 6, 7opelopabb 34421 . 2 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
92, 8bitrd 281 1 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1530   ∈ wcel 2107  Vcvv 3493  ⟨cop 4565  {copab 5119 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-opab 5120 This theorem is referenced by:  bj-opelidb  34431
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