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Theorem rexopabb 5454
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⟩ → (𝜓𝜒))
Assertion
Ref Expression
rexopabb (∃𝑜𝑂 𝜓 ↔ ∃𝑥𝑦(𝜑𝜒))
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
StepHypRef Expression
1 rexopabb.o . . 3 𝑂 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
21rexeqi 3359 . 2 (∃𝑜𝑂 𝜓 ↔ ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
3 elopab 5453 . . . . 5 (𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4 simprr 771 . . . . . . . . 9 ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → 𝜑)
5 rexopabb.p . . . . . . . . . . . 12 (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓𝜒))
65biimpd 228 . . . . . . . . . . 11 (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓𝜒))
76adantr 482 . . . . . . . . . 10 ((𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝜓𝜒))
87impcom 409 . . . . . . . . 9 ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → 𝜒)
94, 8jca 513 . . . . . . . 8 ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → (𝜑𝜒))
109ex 414 . . . . . . 7 (𝜓 → ((𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝜑𝜒)))
11102eximdv 1920 . . . . . 6 (𝜓 → (∃𝑥𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥𝑦(𝜑𝜒)))
1211impcom 409 . . . . 5 ((∃𝑥𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∧ 𝜓) → ∃𝑥𝑦(𝜑𝜒))
133, 12sylanb 582 . . . 4 ((𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝜓) → ∃𝑥𝑦(𝜑𝜒))
1413rexlimiva 3141 . . 3 (∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓 → ∃𝑥𝑦(𝜑𝜒))
15 nfopab1 5151 . . . . 5 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
16 nfv 1915 . . . . 5 𝑥𝜓
1715, 16nfrex 3301 . . . 4 𝑥𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓
18 nfopab2 5152 . . . . . 6 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
19 nfv 1915 . . . . . 6 𝑦𝜓
2018, 19nfrex 3301 . . . . 5 𝑦𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓
21 opabidw 5450 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
22 opex 5392 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
2322, 5sbcie 3764 . . . . . 6 ([𝑥, 𝑦⟩ / 𝑜]𝜓𝜒)
24 rspesbca 3819 . . . . . 6 ((⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ [𝑥, 𝑦⟩ / 𝑜]𝜓) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
2521, 23, 24syl2anbr 600 . . . . 5 ((𝜑𝜒) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
2620, 25exlimi 2208 . . . 4 (∃𝑦(𝜑𝜒) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
2717, 26exlimi 2208 . . 3 (∃𝑥𝑦(𝜑𝜒) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
2814, 27impbii 208 . 2 (∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓 ↔ ∃𝑥𝑦(𝜑𝜒))
292, 28bitri 275 1 (∃𝑜𝑂 𝜓 ↔ ∃𝑥𝑦(𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1539  wex 1779  wcel 2104  wrex 3071  [wsbc 3721  cop 4571  {copab 5143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-sbc 3722  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-opab 5144
This theorem is referenced by:  satfv1  33370
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