Theorem opelopabd 34436

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 5414	. 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 34434	. 2 ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
10	1, 9	syl5bb 285	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780  Ⅎwnf 1784   ∈ wcel 2114  ⟨cop 4573  {copab 5128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5129

This theorem is referenced by:  brabd0  34442