Theorem opelopabd 35239

Description: Membership of an ordere pair in a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)

Hypotheses

Ref	Expression
opelopabd.xph	⊢ (𝜑 → ∀𝑥𝜑)
opelopabd.yph	⊢ (𝜑 → ∀𝑦𝜑)
opelopabd.xch	⊢ (𝜑 → Ⅎ𝑥𝜒)
opelopabd.ych	⊢ (𝜑 → Ⅎ𝑦𝜒)
opelopabd.exa	⊢ (𝜑 → 𝐴 ∈ 𝑈)
opelopabd.exb	⊢ (𝜑 → 𝐵 ∈ 𝑉)
opelopabd.is	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
opelopabd	⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜒))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem opelopabd

Step	Hyp	Ref	Expression
1		elopab 5433	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
2		opelopabd.xph	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3		opelopabd.yph	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
4		opelopabd.xch	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒)
5		opelopabd.ych	. . 3 ⊢ (𝜑 → Ⅎ𝑦𝜒)
6		opelopabd.exa	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
7		opelopabd.exb	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
8		opelopabd.is	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
9	2, 3, 4, 5, 6, 7, 8	copsex2d 35237	. 2 ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
10	1, 9	syl5bb 282	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539  ∃wex 1783  Ⅎwnf 1787   ∈ wcel 2108  ⟨cop 4564  {copab 5132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133

This theorem is referenced by:  brabd0  35245