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Theorem ralopabb 41757
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 1831 . . 3 (¬ ∀𝑥𝑦(𝜑𝜒) ↔ ∃𝑥𝑦 ¬ (𝜑𝜒))
2 ralopabb.o . . . . 5 𝑂 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
3 ralopabb.p . . . . . 6 (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓𝜒))
43notbid 318 . . . . 5 (𝑜 = ⟨𝑥, 𝑦⟩ → (¬ 𝜓 ↔ ¬ 𝜒))
52, 4rexopabb 5490 . . . 4 (∃𝑜𝑂 ¬ 𝜓 ↔ ∃𝑥𝑦(𝜑 ∧ ¬ 𝜒))
6 annim 405 . . . . 5 ((𝜑 ∧ ¬ 𝜒) ↔ ¬ (𝜑𝜒))
762exbii 1852 . . . 4 (∃𝑥𝑦(𝜑 ∧ ¬ 𝜒) ↔ ∃𝑥𝑦 ¬ (𝜑𝜒))
85, 7bitri 275 . . 3 (∃𝑜𝑂 ¬ 𝜓 ↔ ∃𝑥𝑦 ¬ (𝜑𝜒))
9 rexnal 3104 . . 3 (∃𝑜𝑂 ¬ 𝜓 ↔ ¬ ∀𝑜𝑂 𝜓)
101, 8, 93bitr2ri 300 . 2 (¬ ∀𝑜𝑂 𝜓 ↔ ¬ ∀𝑥𝑦(𝜑𝜒))
1110con4bii 321 1 (∀𝑜𝑂 𝜓 ↔ ∀𝑥𝑦(𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wex 1782  wral 3065  wrex 3074  cop 4597  {copab 5172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5173
This theorem is referenced by: (None)
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