Theorem opabidw 5494

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 5495 with a disjoint variable condition, which does not require ax-13 2403. (Contributed by NM, 14-Apr-1995.) Avoid ax-13 2403. (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 5431	. 2 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
2		copsexgw 5458	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3	2	bicomd 225	. 2 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑))
4		df-opab 5163	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
5	1, 3, 4	elab2 3641	1 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1560  ∃wex 1799   ∈ wcel 2142  ⟨cop 4588  {copab 5162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163

This theorem is referenced by:  rexopabb  5498  ssopab2bw  5518  dmopab  5891  rnopab  5930  funopab  6556  opabiota  6949  fvopab5  7009  f1ompt  7092  ovid  7537  zfrep6OLD  7936  enssdomOLD  8958  omxpenlem  9050  infxpenlem  9969  canthwelem  10608  pospo  18375  2ndcdisj  23516  lgsquadlem1  27444  lgsquadlem2  27445  h2hlm  31183  opabdm  32813  opabrn  32814  fpwrelmap  32935  eulerpartlemgvv  34673  fineqvrep  35410  satfvsucsuc  35715  bj-opelopabid  37679  phpreu  38103  poimirlem26  38145  vvdifopab  38764  brabidgaw  38872  diclspsn  41818  areaquad  43793  sprsymrelf  48101