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Theorem opelopab4 38246
 Description: Ordered pair membership in a class abstraction of pairs. Compare to elopab 4943. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
opelopab4 (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣   𝑦,𝑣
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem opelopab4
StepHypRef Expression
1 elopab 4943 . 2 (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 vex 3189 . . . . . 6 𝑥 ∈ V
3 vex 3189 . . . . . 6 𝑦 ∈ V
42, 3opth 4905 . . . . 5 (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ ↔ (𝑥 = 𝑢𝑦 = 𝑣))
5 eqcom 2628 . . . . 5 (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ ↔ ⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
64, 5bitr3i 266 . . . 4 ((𝑥 = 𝑢𝑦 = 𝑣) ↔ ⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
76anbi1i 730 . . 3 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ↔ (⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
872exbii 1772 . 2 (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ↔ ∃𝑥𝑦(⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
91, 8bitr4i 267 1 (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ⟨cop 4154  {copab 4672 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-opab 4674 This theorem is referenced by: (None)
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