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Theorem opelopabt 4912
 Description: Closed theorem form of opelopab 4922. (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 4908 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 19.26-2 1787 . . . 4 (∀𝑥𝑦((𝑥 = 𝐴 → (𝜑𝜓)) ∧ (𝑦 = 𝐵 → (𝜓𝜒))) ↔ (∀𝑥𝑦(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝑦(𝑦 = 𝐵 → (𝜓𝜒))))
3 prth 593 . . . . . 6 (((𝑥 = 𝐴 → (𝜑𝜓)) ∧ (𝑦 = 𝐵 → (𝜓𝜒))) → ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝜑𝜓) ∧ (𝜓𝜒))))
4 bitr 741 . . . . . 6 (((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))
53, 4syl6 34 . . . . 5 (((𝑥 = 𝐴 → (𝜑𝜓)) ∧ (𝑦 = 𝐵 → (𝜓𝜒))) → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒)))
652alimi 1731 . . . 4 (∀𝑥𝑦((𝑥 = 𝐴 → (𝜑𝜓)) ∧ (𝑦 = 𝐵 → (𝜓𝜒))) → ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒)))
72, 6sylbir 224 . . 3 ((∀𝑥𝑦(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝑦(𝑦 = 𝐵 → (𝜓𝜒))) → ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒)))
8 copsex2t 4883 . . 3 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒)) ∧ (𝐴𝑉𝐵𝑊)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜒))
97, 8stoic3 1692 . 2 ((∀𝑥𝑦(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝑦(𝑦 = 𝐵 → (𝜓𝜒)) ∧ (𝐴𝑉𝐵𝑊)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜒))
101, 9syl5bb 271 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 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ⟨cop 4131  {copab 4642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644 This theorem is referenced by: (None)
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