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Theorem opabid 5535
Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Usage of this theorem is discouraged because it depends on ax-13 2375. Use the weaker opabidw 5534 when possible. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.)
Assertion
Ref Expression
opabid (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)

Proof of Theorem opabid
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 opex 5475 . 2 𝑥, 𝑦⟩ ∈ V
2 copsexg 5502 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
32bicomd 223 . 2 (𝑧 = ⟨𝑥, 𝑦⟩ → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑))
4 df-opab 5211 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
51, 3, 4elab2 3685 1 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  cop 4637  {copab 5210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-13 2375  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211
This theorem is referenced by:  ssopab2b  5559  brabidga  38348
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