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Theorem ralopabb 41997
Description: Restricted universal quantification over an ordered-pair class abstraction. (Contributed by RP, 25-Sep-2024.)
Hypotheses
Ref Expression
ralopabb.o 𝑂 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
ralopabb.p (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓𝜒))
Assertion
Ref Expression
ralopabb (∀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 ralopabb
StepHypRef Expression
1 2nalexn 1830 . . 3 (¬ ∀𝑥𝑦(𝜑𝜒) ↔ ∃𝑥𝑦 ¬ (𝜑𝜒))
2 ralopabb.o . . . . 5 𝑂 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
3 ralopabb.p . . . . . 6 (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓𝜒))
43notbid 317 . . . . 5 (𝑜 = ⟨𝑥, 𝑦⟩ → (¬ 𝜓 ↔ ¬ 𝜒))
52, 4rexopabb 5522 . . . 4 (∃𝑜𝑂 ¬ 𝜓 ↔ ∃𝑥𝑦(𝜑 ∧ ¬ 𝜒))
6 annim 404 . . . . 5 ((𝜑 ∧ ¬ 𝜒) ↔ ¬ (𝜑𝜒))
762exbii 1851 . . . 4 (∃𝑥𝑦(𝜑 ∧ ¬ 𝜒) ↔ ∃𝑥𝑦 ¬ (𝜑𝜒))
85, 7bitri 274 . . 3 (∃𝑜𝑂 ¬ 𝜓 ↔ ∃𝑥𝑦 ¬ (𝜑𝜒))
9 rexnal 3100 . . 3 (∃𝑜𝑂 ¬ 𝜓 ↔ ¬ ∀𝑜𝑂 𝜓)
101, 8, 93bitr2ri 299 . 2 (¬ ∀𝑜𝑂 𝜓 ↔ ¬ ∀𝑥𝑦(𝜑𝜒))
1110con4bii 320 1 (∀𝑜𝑂 𝜓 ↔ ∀𝑥𝑦(𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wex 1781  wral 3061  wrex 3070  cop 4629  {copab 5204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3775  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5205
This theorem is referenced by: (None)
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