Theorem rexopabb 5547

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 3333	. 2 ⊢ (∃𝑜 ∈ 𝑂 𝜓 ↔ ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
3		elopab 5546	. . . . 5 ⊢ (𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		simprr 772	. . . . . . . . 9 ⊢ ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → 𝜑)
5		rexopabb.p	. . . . . . . . . . . 12 ⊢ (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜒))
6	5	biimpd 229	. . . . . . . . . . 11 ⊢ (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓 → 𝜒))
7	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝜓 → 𝜒))
8	7	impcom 407	. . . . . . . . 9 ⊢ ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → 𝜒)
9	4, 8	jca 511	. . . . . . . 8 ⊢ ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → (𝜑 ∧ 𝜒))
10	9	ex 412	. . . . . . 7 ⊢ (𝜓 → ((𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝜑 ∧ 𝜒)))
11	10	2eximdv 1918	. . . . . 6 ⊢ (𝜓 → (∃𝑥∃𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦(𝜑 ∧ 𝜒)))
12	11	impcom 407	. . . . 5 ⊢ ((∃𝑥∃𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∧ 𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
13	3, 12	sylanb 580	. . . 4 ⊢ ((𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
14	13	rexlimiva 3153	. . 3 ⊢ (∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓 → ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
15		nfopab1 5236	. . . . 5 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
16		nfv 1913	. . . . 5 ⊢ Ⅎ𝑥𝜓
17	15, 16	nfrexw 3319	. . . 4 ⊢ Ⅎ𝑥∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓
18		nfopab2 5237	. . . . . 6 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
19		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑦𝜓
20	18, 19	nfrexw 3319	. . . . 5 ⊢ Ⅎ𝑦∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓
21		opabidw 5543	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
22		opex 5484	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
23	22, 5	sbcie 3848	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩ / 𝑜]𝜓 ↔ 𝜒)
24		rspesbca 3903	. . . . . 6 ⊢ ((⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ [⟨𝑥, 𝑦⟩ / 𝑜]𝜓) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
25	21, 23, 24	syl2anbr 598	. . . . 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 209	. 2 ⊢ (∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓 ↔ ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
29	2, 28	bitri 275	1 ⊢ (∃𝑜 ∈ 𝑂 𝜓 ↔ ∃𝑥∃𝑦(𝜑 ∧ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076  [wsbc 3804  ⟨cop 4654  {copab 5228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229

This theorem is referenced by:  satfv1  35331  ralopabb  43373