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Theorem rexopabb 5411
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 3419 . 2 (∃𝑜𝑂 𝜓 ↔ ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
3 elopab 5410 . . . . 5 (𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4 simprr 769 . . . . . . . . 9 ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → 𝜑)
5 rexopabb.p . . . . . . . . . . . 12 (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓𝜒))
65biimpd 230 . . . . . . . . . . 11 (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓𝜒))
76adantr 481 . . . . . . . . . 10 ((𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝜓𝜒))
87impcom 408 . . . . . . . . 9 ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → 𝜒)
94, 8jca 512 . . . . . . . 8 ((𝜓 ∧ (𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) → (𝜑𝜒))
109ex 413 . . . . . . 7 (𝜓 → ((𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝜑𝜒)))
11102eximdv 1913 . . . . . 6 (𝜓 → (∃𝑥𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥𝑦(𝜑𝜒)))
1211impcom 408 . . . . 5 ((∃𝑥𝑦(𝑜 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∧ 𝜓) → ∃𝑥𝑦(𝜑𝜒))
133, 12sylanb 581 . . . 4 ((𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝜓) → ∃𝑥𝑦(𝜑𝜒))
1413rexlimiva 3285 . . 3 (∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓 → ∃𝑥𝑦(𝜑𝜒))
15 nfopab1 5131 . . . . 5 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
16 nfv 1908 . . . . 5 𝑥𝜓
1715, 16nfrex 3313 . . . 4 𝑥𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓
18 nfopab2 5132 . . . . . 6 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
19 nfv 1908 . . . . . 6 𝑦𝜓
2018, 19nfrex 3313 . . . . 5 𝑦𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓
21 opabidw 5408 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
22 opex 5352 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
2322, 5sbcie 3815 . . . . . 6 ([𝑥, 𝑦⟩ / 𝑜]𝜓𝜒)
24 rspesbca 3867 . . . . . 6 ((⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ [𝑥, 𝑦⟩ / 𝑜]𝜓) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
2521, 23, 24syl2anbr 598 . . . . 5 ((𝜑𝜒) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
2620, 25exlimi 2210 . . . 4 (∃𝑦(𝜑𝜒) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
2717, 26exlimi 2210 . . 3 (∃𝑥𝑦(𝜑𝜒) → ∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓)
2814, 27impbii 210 . 2 (∃𝑜 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}𝜓 ↔ ∃𝑥𝑦(𝜑𝜒))
292, 28bitri 276 1 (∃𝑜𝑂 𝜓 ↔ ∃𝑥𝑦(𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  wrex 3143  [wsbc 3775  cop 4569  {copab 5124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-opab 5125
This theorem is referenced by:  satfv1  32497
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