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Theorem opelopab2a 5497
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 2826 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐶𝐴𝐶))
2 eleq1 2826 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐷𝐵𝐷))
31, 2bi2anan9 638 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝐶𝑦𝐷) ↔ (𝐴𝐶𝐵𝐷)))
4 opelopabga.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
53, 4anbi12d 632 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (((𝑥𝐶𝑦𝐷) ∧ 𝜑) ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓)))
65opelopabga 5495 . 2 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓)))
76bianabs 543 1 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cop 4597  {copab 5172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5173
This theorem is referenced by:  opelopab2  5503  brab2a  5730  prdsleval  17366  isperp  27696  rrx2plord  46880  rrx2plordisom  46883
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