Theorem opelopabd 37498

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 5472	. 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 37496	. 2 ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
10	1, 9	bitrid 284	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1541   = wceq 1543  ∃wex 1782  Ⅎwnf 1786   ∈ wcel 2115  ⟨cop 4564  {copab 5137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-10 2148  ax-11 2164  ax-12 2185  ax-ext 2708  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-3an 1090  df-tru 1546  df-fal 1556  df-ex 1783  df-nf 1787  df-sb 2070  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3389  df-v 3430  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5138

This theorem is referenced by:  brabd0  37504