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Theorem opelopabt 5183
Description: Closed theorem form of opelopab 5193. (Contributed by NM, 19-Feb-2013.)
Assertion
Ref Expression
opelopabt ((∀𝑥𝑦(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝑦(𝑦 = 𝐵 → (𝜓𝜒)) ∧ (𝐴𝑉𝐵𝑊)) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opelopabt
StepHypRef Expression
1 elopab 5179 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 19.26-2 1970 . . . 4 (∀𝑥𝑦((𝑥 = 𝐴 → (𝜑𝜓)) ∧ (𝑦 = 𝐵 → (𝜓𝜒))) ↔ (∀𝑥𝑦(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝑦(𝑦 = 𝐵 → (𝜓𝜒))))
3 prth 844 . . . . . 6 (((𝑥 = 𝐴 → (𝜑𝜓)) ∧ (𝑦 = 𝐵 → (𝜓𝜒))) → ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝜑𝜓) ∧ (𝜓𝜒))))
4 bitr 840 . . . . . 6 (((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))
53, 4syl6 35 . . . . 5 (((𝑥 = 𝐴 → (𝜑𝜓)) ∧ (𝑦 = 𝐵 → (𝜓𝜒))) → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒)))
652alimi 1908 . . . 4 (∀𝑥𝑦((𝑥 = 𝐴 → (𝜑𝜓)) ∧ (𝑦 = 𝐵 → (𝜓𝜒))) → ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒)))
72, 6sylbir 227 . . 3 ((∀𝑥𝑦(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝑦(𝑦 = 𝐵 → (𝜓𝜒))) → ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒)))
8 copsex2t 5147 . . 3 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒)) ∧ (𝐴𝑉𝐵𝑊)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜒))
97, 8stoic3 1872 . 2 ((∀𝑥𝑦(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝑦(𝑦 = 𝐵 → (𝜓𝜒)) ∧ (𝐴𝑉𝐵𝑊)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜒))
101, 9syl5bb 275 1 ((∀𝑥𝑦(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝑦(𝑦 = 𝐵 → (𝜓𝜒)) ∧ (𝐴𝑉𝐵𝑊)) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108  wal 1651   = wceq 1653  wex 1875  wcel 2157  cop 4374  {copab 4905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-opab 4906
This theorem is referenced by: (None)
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