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Theorem copsex2t 5497
Description: Closed theorem form of copsex2g 5498. (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 2151 . . 3 𝑥𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
2 nfe1 2150 . . . 4 𝑥𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
3 nfv 1914 . . . 4 𝑥𝜓
42, 3nfbi 1903 . . 3 𝑥(∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)
5 nfa2 2176 . . . 4 𝑦𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
6 nfe1 2150 . . . . . 6 𝑦𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
76nfex 2324 . . . . 5 𝑦𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
8 nfv 1914 . . . . 5 𝑦𝜓
97, 8nfbi 1903 . . . 4 𝑦(∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)
10 opeq12 4875 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
11 copsexgw 5495 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
1211eqcoms 2745 . . . . . . . 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 481 . . . . . 6 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜑 ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
15 2sp 2186 . . . . . . 7 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)))
1615imp 406 . . . . . 6 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜑𝜓))
1714, 16bitr3d 281 . . . . 5 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
1817ex 412 . . . 4 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) → ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)))
195, 9, 18exlimd 2218 . . 3 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) → (∃𝑦(𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)))
201, 4, 19exlimd 2218 . 2 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) → (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)))
21 elisset 2823 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
22 elisset 2823 . . . 4 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
2321, 22anim12i 613 . . 3 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
24 exdistrv 1955 . . 3 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
2523, 24sylibr 234 . 2 ((𝐴𝑉𝐵𝑊) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
2620, 25impel 505 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 206  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  cop 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633
This theorem is referenced by:  copsex2dv  5499  opelopabt  5537
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