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Theorem opelopab2a 4250
Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypothesis
Ref Expression
opelopabga.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
opelopab2a ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ 𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opelopab2a
StepHypRef Expression
1 eleq1 2233 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐶𝐴𝐶))
2 eleq1 2233 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐷𝐵𝐷))
31, 2bi2anan9 601 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝐶𝑦𝐷) ↔ (𝐴𝐶𝐵𝐷)))
4 opelopabga.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
53, 4anbi12d 470 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (((𝑥𝐶𝑦𝐷) ∧ 𝜑) ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓)))
65opelopabga 4248 . 2 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓)))
76bianabs 606 1 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  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:  opelopab2  4255  brab2a  4664  brab2ga  4686  ltdfpr  7468
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