Theorem rexopabb 5415

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 3414	. 2 ⊢ (∃𝑜 ∈ 𝑂 𝜓 ↔ ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
3		elopab 5414	. . . . 5 ⊢ (𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4		simprr 771	. . . . . . . . 9 ⊢ ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → 𝜑)
5		rexopabb.p	. . . . . . . . . . . 12 ⊢ (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜒))
6	5	biimpd 231	. . . . . . . . . . 11 ⊢ (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓 → 𝜒))
7	6	adantr 483	. . . . . . . . . 10 ⊢ ((𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝜓 → 𝜒))
8	7	impcom 410	. . . . . . . . 9 ⊢ ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → 𝜒)
9	4, 8	jca 514	. . . . . . . 8 ⊢ ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → (𝜑 ∧ 𝜒))
10	9	ex 415	. . . . . . 7 ⊢ (𝜓 → ((𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝜑 ∧ 𝜒)))
11	10	2eximdv 1920	. . . . . 6 ⊢ (𝜓 → (∃𝑥∃𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦(𝜑 ∧ 𝜒)))
12	11	impcom 410	. . . . 5 ⊢ ((∃𝑥∃𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∧ 𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
13	3, 12	sylanb 583	. . . 4 ⊢ ((𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
14	13	rexlimiva 3281	. . 3 ⊢ (∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓 → ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
15		nfopab1 5135	. . . . 5 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
16		nfv 1915	. . . . 5 ⊢ Ⅎ𝑥𝜓
17	15, 16	nfrex 3309	. . . 4 ⊢ Ⅎ𝑥∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓
18		nfopab2 5136	. . . . . 6 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
19		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑦𝜓
20	18, 19	nfrex 3309	. . . . 5 ⊢ Ⅎ𝑦∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓
21		opabidw 5412	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
22		opex 5356	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
23	22, 5	sbcie 3812	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩ / 𝑜]𝜓 ↔ 𝜒)
24		rspesbca 3864	. . . . . 6 ⊢ ((⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ [⟨𝑥, 𝑦⟩ / 𝑜]𝜓) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
25	21, 23, 24	syl2anbr 600	. . . . 5 ⊢ ((𝜑 ∧ 𝜒) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
26	20, 25	exlimi 2217	. . . 4 ⊢ (∃𝑦(𝜑 ∧ 𝜒) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
27	17, 26	exlimi 2217	. . 3 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜒) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
28	14, 27	impbii 211	. 2 ⊢ (∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓 ↔ ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
29	2, 28	bitri 277	1 ⊢ (∃𝑜 ∈ 𝑂 𝜓 ↔ ∃𝑥∃𝑦(𝜑 ∧ 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∃wrex 3139  [wsbc 3772  ⟨cop 4573  {copab 5128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5129

This theorem is referenced by:  satfv1  32610