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Theorem opelopabt 4195
 Description: Closed theorem form of opelopab 4204. (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 4191 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 19.26-2 1459 . . . . 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 464 . . . . . . 7 (((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))
53, 4syl6 33 . . . . . 6 (((𝑥 = 𝐴 → (𝜑𝜓)) ∧ (𝑦 = 𝐵 → (𝜓𝜒))) → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜒)))
652alimi 1433 . . . . 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 4178 . . . 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 1177 . 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 963  ∀wal 1330   = wceq 1332  ∃wex 1469   ∈ wcel 2112  ⟨cop 3537  {copab 3998 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1732  df-eu 1993  df-mo 1994  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-v 2693  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-opab 4000 This theorem is referenced by: (None)
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