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Theorem ralopabb 42738
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 1822 . . 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 5521 . . . 4 (∃𝑜𝑂 ¬ 𝜓 ↔ ∃𝑥𝑦(𝜑 ∧ ¬ 𝜒))
6 annim 403 . . . . 5 ((𝜑 ∧ ¬ 𝜒) ↔ ¬ (𝜑𝜒))
762exbii 1843 . . . 4 (∃𝑥𝑦(𝜑 ∧ ¬ 𝜒) ↔ ∃𝑥𝑦 ¬ (𝜑𝜒))
85, 7bitri 275 . . 3 (∃𝑜𝑂 ¬ 𝜓 ↔ ∃𝑥𝑦 ¬ (𝜑𝜒))
9 rexnal 3094 . . 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 395  wal 1531   = wceq 1533  wex 1773  wral 3055  wrex 3064  cop 4629  {copab 5203
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5204
This theorem is referenced by: (None)
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