Theorem opabidw 5543

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 5544 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 14-Apr-1995.) Avoid ax-13 2380. (Revised by GG, 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 5484	. 2 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
2		copsexgw 5510	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3	2	bicomd 223	. 2 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑))
4		df-opab 5229	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
5	1, 3, 4	elab2 3698	1 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ⟨cop 4654  {copab 5228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229

This theorem is referenced by:  rexopabb  5547  ssopab2bw  5566  dmopab  5940  rnopab  5979  funopab  6613  opabiota  7004  fvopab5  7062  f1ompt  7145  ovid  7591  zfrep6  7995  enssdom  9037  omxpenlem  9139  infxpenlem  10082  canthwelem  10719  pospo  18415  2ndcdisj  23485  lgsquadlem1  27442  lgsquadlem2  27443  h2hlm  31012  opabdm  32633  opabrn  32634  fpwrelmap  32747  eulerpartlemgvv  34341  fineqvrep  35071  satfvsucsuc  35333  bj-opelopabid  37153  phpreu  37564  poimirlem26  37606  vvdifopab  38216  brabidgaw  38321  diclspsn  41151  areaquad  43177  sprsymrelf  47369