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Theorem opelopabt 4247
Description: Closed theorem form of opelopab 4256. (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 4243 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 19.26-2 1475 . . . . 5 (∀𝑥𝑦((𝑥 = 𝐴 → (𝜑𝜓)) ∧ (𝑦 = 𝐵 → (𝜓𝜒))) ↔ (∀𝑥𝑦(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝑦(𝑦 = 𝐵 → (𝜓𝜒))))
3 anim12 342 . . . . . . 7 (((𝑥 = 𝐴 → (𝜑𝜓)) ∧ (𝑦 = 𝐵 → (𝜓𝜒))) → ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝜑𝜓) ∧ (𝜓𝜒))))
4 bitr 469 . . . . . . 7 (((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))
53, 4syl6 33 . . . . . 6 (((𝑥 = 𝐴 → (𝜑𝜓)) ∧ (𝑦 = 𝐵 → (𝜓𝜒))) → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒)))
652alimi 1449 . . . . 5 (∀𝑥𝑦((𝑥 = 𝐴 → (𝜑𝜓)) ∧ (𝑦 = 𝐵 → (𝜓𝜒))) → ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒)))
72, 6sylbir 134 . . . 4 ((∀𝑥𝑦(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝑦(𝑦 = 𝐵 → (𝜓𝜒))) → ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒)))
8 copsex2t 4230 . . . 4 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒)) ∧ (𝐴𝑉𝐵𝑊)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜒))
97, 8sylan 281 . . 3 (((∀𝑥𝑦(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝑦(𝑦 = 𝐵 → (𝜓𝜒))) ∧ (𝐴𝑉𝐵𝑊)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜒))
1093impa 1189 . 2 ((∀𝑥𝑦(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝑦(𝑦 = 𝐵 → (𝜓𝜒)) ∧ (𝐴𝑉𝐵𝑊)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜒))
111, 10syl5bb 191 1 ((∀𝑥𝑦(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝑦(𝑦 = 𝐵 → (𝜓𝜒)) ∧ (𝐴𝑉𝐵𝑊)) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 973  wal 1346   = wceq 1348  wex 1485  wcel 2141  cop 3586  {copab 4049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051
This theorem is referenced by: (None)
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