Theorem rexopabb 5502

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 3319	. 2 ⊢ (∃𝑜 ∈ 𝑂 𝜓 ↔ ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
3		elopab 5501	. . . . 5 ⊢ (𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		simprr 785	. . . . . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝜓 → 𝜒))
8	7	impcom 413	. . . . . . . . 9 ⊢ ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → 𝜒)
9	4, 8	jca 521	. . . . . . . 8 ⊢ ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → (𝜑 ∧ 𝜒))
10	9	ex 418	. . . . . . 7 ⊢ (𝜓 → ((𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝜑 ∧ 𝜒)))
11	10	2eximdv 1952	. . . . . 6 ⊢ (𝜓 → (∃𝑥∃𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦(𝜑 ∧ 𝜒)))
12	11	impcom 413	. . . . 5 ⊢ ((∃𝑥∃𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∧ 𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
13	3, 12	sylanb 593	. . . 4 ⊢ ((𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
14	13	rexlimiva 3156	. . 3 ⊢ (∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓 → ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
15		nfopab1 5175	. . . . 5 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
16		nfv 1947	. . . . 5 ⊢ Ⅎ𝑥𝜓
17	15, 16	nfrexw 3311	. . . 4 ⊢ Ⅎ𝑥∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓
18		nfopab2 5176	. . . . . 6 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
19		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑦𝜓
20	18, 19	nfrexw 3311	. . . . 5 ⊢ Ⅎ𝑦∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓
21		opabidw 5498	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
22		opex 5432	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
23	22, 5	sbcie 3780	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩ / 𝑜]𝜓 ↔ 𝜒)
24		rspesbca 3828	. . . . . 6 ⊢ ((⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ [⟨𝑥, 𝑦⟩ / 𝑜]𝜓) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
25	21, 23, 24	syl2anbr 611	. . . . 5 ⊢ ((𝜑 ∧ 𝜒) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
26	20, 25	exlimi 2254	. . . 4 ⊢ (∃𝑦(𝜑 ∧ 𝜒) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
27	17, 26	exlimi 2254	. . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3087  [wsbc 3739  ⟨cop 4590  {copab 5167

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5168

This theorem is used by:  satfv1  36097  ralopabb  44370