Theorem opabidw 5510

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 5511 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by NM, 14-Apr-1995.) Avoid ax-13 2406. (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 5447	. 2 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
2		copsexgw 5474	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3	2	bicomd 226	. 2 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑))
4		df-opab 5176	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
5	1, 3, 4	elab2 3643	1 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ⟨cop 4597  {copab 5175

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5176

This theorem is used by:  rexopabb  5514  ssopab2bw  5534  dmopab  5907  rnopab  5946  funopab  6575  opabiota  6967  fvopab5  7027  f1ompt  7110  ovid  7560  zfrep6OLD  7958  enssdomOLD  8980  omxpenlem  9073  infxpenlem  10013  canthwelem  10652  pospo  18423  2ndcdisj  23666  lgsquadlem1  27597  lgsquadlem2  27598  h2hlm  31405  opabdm  33029  opabrn  33030  fpwrelmap  33150  eulerpartlemgvv  34833  fineqvrep  35586  satfvsucsuc  35896  bj-opelopabid  37890  phpreu  38314  poimirlem26  38356  vvdifopab  38974  brabidgaw  39082  diclspsn  42028  areaquad  44003  sprsymrelf  48304