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Theorem copsex2t 4955
Description: Closed theorem form of copsex2g 4956. (Contributed by NM, 17-Feb-2013.)
Assertion
Ref Expression
copsex2t ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) ∧ (𝐴𝑉𝐵𝑊)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝑦,𝜓   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem copsex2t
StepHypRef Expression
1 nfa1 2027 . . 3 𝑥𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
2 nfe1 2026 . . . 4 𝑥𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
3 nfv 1842 . . . 4 𝑥𝜓
42, 3nfbi 1832 . . 3 𝑥(∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)
5 nfa2 2039 . . . 4 𝑦𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
6 nfe1 2026 . . . . . 6 𝑦𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
76nfex 2153 . . . . 5 𝑦𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
8 nfv 1842 . . . . 5 𝑦𝜓
97, 8nfbi 1832 . . . 4 𝑦(∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)
10 opeq12 4402 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
11 copsexg 4954 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
1211eqcoms 2629 . . . . . . . 8 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → (𝜑 ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
1310, 12syl 17 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑 ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
1413adantl 482 . . . . . 6 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜑 ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
15 2sp 2055 . . . . . . 7 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)))
1615imp 445 . . . . . 6 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜑𝜓))
1714, 16bitr3d 270 . . . . 5 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
1817ex 450 . . . 4 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) → ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)))
195, 9, 18exlimd 2086 . . 3 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) → (∃𝑦(𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)))
201, 4, 19exlimd 2086 . 2 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) → (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)))
21 elisset 3213 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
22 elisset 3213 . . . 4 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
2321, 22anim12i 590 . . 3 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
24 eeanv 2181 . . 3 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
2523, 24sylibr 224 . 2 ((𝐴𝑉𝐵𝑊) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
2620, 25impel 485 1 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) ∧ (𝐴𝑉𝐵𝑊)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1480   = wceq 1482  wex 1703  wcel 1989  cop 4181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182
This theorem is referenced by:  opelopabt  4985
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