Theorem opabidw 5505

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 5506 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by NM, 14-Apr-1995.) Avoid ax-13 2404. (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 5439	. 2 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
2		copsexgw 5467	. . 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 5174	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
5	1, 3, 4	elab2 3639	1 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ⟨cop 4840  {copab 5173

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-opab 5174

This theorem is used by:  rexopabb  5509  ssopab2bw  5529  dmopab  5909  rnopab  5948  funopab  6603  opabiota  6998  fvopab5  7058  f1ompt  7143  ovid  7593  zfrep6OLD  8001  enssdomOLD  9042  omxpenlem  9135  infxpenlem  10131  canthwelem  10774  pospo  18567  2ndcdisj  23830  lgsquadlem1  27764  lgsquadlem2  27765  h2hlm  31639  opabdm  33303  opabrn  33304  fpwrelmap  33421  eulerpartlemgvv  35119  fwfffun  35828  rankse  35864  fineqvrep  35923  satfvsucsuc  36338  bj-opelopabid  38376  phpreu  38795  poimirlem26  38832  vvdifopab  39463  brabidgaw  39571  diclspsn  42517  areaquad  44479  sprsymrelf  48850