Theorem opelopabb 37639

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

Hypotheses

Ref	Expression
opelopabb.xph	⊢ (𝜑 → ∀𝑥𝜑)
opelopabb.yph	⊢ (𝜑 → ∀𝑦𝜑)
opelopabb.xch	⊢ (𝜑 → Ⅎ𝑥𝜒)
opelopabb.ych	⊢ (𝜑 → Ⅎ𝑦𝜒)
opelopabb.is	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))

Assertion

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

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

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

Proof of Theorem opelopabb

Step	Hyp	Ref	Expression
1		elopab 5498	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
2		opelopabb.xph	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3		opelopabb.yph	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
4		opelopabb.xch	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒)
5		opelopabb.ych	. . 3 ⊢ (𝜑 → Ⅎ𝑦𝜒)
6		opelopabb.is	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
7	2, 3, 4, 5, 6	copsex2b 37637	. 2 ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
8	1, 7	bitrid 285	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1559   = wceq 1561  ∃wex 1800  Ⅎwnf 1804   ∈ wcel 2143  Vcvv 3455  ⟨cop 4589  {copab 5163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-opab 5164

This theorem is referenced by:  opelopabbv  37640