Theorem opabidw 5526

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 5527 with a disjoint variable condition, which does not require ax-13 2367. (Contributed by NM, 14-Apr-1995.) Avoid ax-13 2367. (Revised by Gino Giotto, 26-Jan-2024.)

Assertion

Ref	Expression
opabidw	⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabidw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 5466	. 2 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
2		copsexgw 5492	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3	2	bicomd 222	. 2 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑))
4		df-opab 5211	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
5	1, 3, 4	elab2 3671	1 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534  ∃wex 1774   ∈ wcel 2099  ⟨cop 4635  {copab 5210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5211

This theorem is referenced by:  rexopabb  5530  ssopab2bw  5549  dmopab  5918  rnopab  5956  funopab  6588  opabiota  6981  fvopab5  7038  f1ompt  7121  ovid  7562  zfrep6  7958  enssdom  8998  omxpenlem  9098  infxpenlem  10037  canthwelem  10674  pospo  18337  2ndcdisj  23373  lgsquadlem1  27326  lgsquadlem2  27327  h2hlm  30803  opabdm  32414  opabrn  32415  fpwrelmap  32528  eulerpartlemgvv  33996  fineqvrep  34715  satfvsucsuc  34975  bj-opelopabid  36666  phpreu  37077  poimirlem26  37119  vvdifopab  37732  brabidgaw  37837  diclspsn  40667  areaquad  42644  sprsymrelf  46835