Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opelopabbv Structured version   Visualization version   GIF version

Theorem opelopabbv 37126
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 2825 . 2 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
3 ax-5 1908 . . 3 (𝜑 → ∀𝑥𝜑)
4 ax-5 1908 . . 3 (𝜑 → ∀𝑦𝜑)
5 nfvd 1913 . . 3 (𝜑 → Ⅎ𝑥𝜒)
6 nfvd 1913 . . 3 (𝜑 → Ⅎ𝑦𝜒)
7 opelopabbv.is . . 3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
83, 4, 5, 6, 7opelopabb 37125 . 2 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
92, 8bitrd 279 1 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637  {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-11 2155  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-opab 5211
This theorem is referenced by:  bj-opelidb  37135
  Copyright terms: Public domain W3C validator