Theorem rexopabb 5380

Description: Restricted existential quantification over an ordered-pair class abstraction. (Contributed by AV, 8-Nov-2023.)

Hypotheses

Ref	Expression
rexopabb.o	⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
rexopabb.p	⊢ (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rexopabb	⊢ (∃𝑜 ∈ 𝑂 𝜓 ↔ ∃𝑥∃𝑦(𝜑 ∧ 𝜒))

Distinct variable groups:   𝑜,𝑂   𝑥,𝑜,𝑦   𝜑,𝑜   𝜓,𝑥,𝑦   𝜒,𝑜

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

Proof of Theorem rexopabb

Step	Hyp	Ref	Expression
1		rexopabb.o	. . 3 ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2	1	rexeqi 3314	. 2 ⊢ (∃𝑜 ∈ 𝑂 𝜓 ↔ ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
3		elopab 5379	. . . . 5 ⊢ (𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		simprr 773	. . . . . . . . 9 ⊢ ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → 𝜑)
5		rexopabb.p	. . . . . . . . . . . 12 ⊢ (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜒))
6	5	biimpd 232	. . . . . . . . . . 11 ⊢ (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓 → 𝜒))
7	6	adantr 484	. . . . . . . . . 10 ⊢ ((𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝜓 → 𝜒))
8	7	impcom 411	. . . . . . . . 9 ⊢ ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → 𝜒)
9	4, 8	jca 515	. . . . . . . 8 ⊢ ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → (𝜑 ∧ 𝜒))
10	9	ex 416	. . . . . . 7 ⊢ (𝜓 → ((𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝜑 ∧ 𝜒)))
11	10	2eximdv 1925	. . . . . 6 ⊢ (𝜓 → (∃𝑥∃𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦(𝜑 ∧ 𝜒)))
12	11	impcom 411	. . . . 5 ⊢ ((∃𝑥∃𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∧ 𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
13	3, 12	sylanb 584	. . . 4 ⊢ ((𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
14	13	rexlimiva 3190	. . 3 ⊢ (∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓 → ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
15		nfopab1 5096	. . . . 5 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
16		nfv 1920	. . . . 5 ⊢ Ⅎ𝑥𝜓
17	15, 16	nfrex 3218	. . . 4 ⊢ Ⅎ𝑥∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓
18		nfopab2 5097	. . . . . 6 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
19		nfv 1920	. . . . . 6 ⊢ Ⅎ𝑦𝜓
20	18, 19	nfrex 3218	. . . . 5 ⊢ Ⅎ𝑦∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓
21		opabidw 5377	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
22		opex 5319	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
23	22, 5	sbcie 3720	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩ / 𝑜]𝜓 ↔ 𝜒)
24		rspesbca 3770	. . . . . 6 ⊢ ((⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ [⟨𝑥, 𝑦⟩ / 𝑜]𝜓) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
25	21, 23, 24	syl2anbr 602	. . . . 5 ⊢ ((𝜑 ∧ 𝜒) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
26	20, 25	exlimi 2218	. . . 4 ⊢ (∃𝑦(𝜑 ∧ 𝜒) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
27	17, 26	exlimi 2218	. . 3 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜒) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
28	14, 27	impbii 212	. 2 ⊢ (∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓 ↔ ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
29	2, 28	bitri 278	1 ⊢ (∃𝑜 ∈ 𝑂 𝜓 ↔ ∃𝑥∃𝑦(𝜑 ∧ 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542  ∃wex 1786   ∈ wcel 2113  ∃wrex 3054  [wsbc 3679  ⟨cop 4519  {copab 5089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-opab 5090

This theorem is referenced by:  satfv1  32888