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Theorem opelopab2a 5395
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 2899 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐶𝐴𝐶))
2 eleq1 2899 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐷𝐵𝐷))
31, 2bi2anan9 638 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝐶𝑦𝐷) ↔ (𝐴𝐶𝐵𝐷)))
4 opelopabga.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
53, 4anbi12d 633 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (((𝑥𝐶𝑦𝐷) ∧ 𝜑) ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓)))
65opelopabga 5393 . 2 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓)))
76bianabs 545 1 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  cop 4546  {copab 5101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-opab 5102
This theorem is referenced by:  opelopab2  5401  brab2a  5617  prdsleval  16728  isperp  26484  rrx2plord  44894  rrx2plordisom  44897
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