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Theorem opelopab4 41191
Description: Ordered pair membership in a class abstraction of pairs. Compare to elopab 5391. (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 5391 . 2 (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 vex 3472 . . . . . 6 𝑥 ∈ V
3 vex 3472 . . . . . 6 𝑦 ∈ V
42, 3opth 5345 . . . . 5 (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ ↔ (𝑥 = 𝑢𝑦 = 𝑣))
5 eqcom 2829 . . . . 5 (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ ↔ ⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
64, 5bitr3i 280 . . . 4 ((𝑥 = 𝑢𝑦 = 𝑣) ↔ ⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
76anbi1i 626 . . 3 (((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ↔ (⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
872exbii 1850 . 2 (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ↔ ∃𝑥𝑦(⟨𝑢, 𝑣⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
91, 8bitr4i 281 1 (⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2114  cop 4545  {copab 5104
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
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 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-opab 5105
This theorem is referenced by: (None)
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