Theorem opelopabd 37571

Description: Membership of an ordered 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 5487	. 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 37569	. 2 ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
10	1, 9	bitrid 285	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1548   = wceq 1550  ∃wex 1789  Ⅎwnf 1793   ∈ wcel 2132  ⟨cop 4578  {copab 5152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-opab 5153

This theorem is referenced by:  brabd0  37577