Theorem opabidw 5473

Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 5474 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 14-Apr-1995.) Avoid ax-13 2377. (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 5412	. 2 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
2		copsexgw 5439	. . 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 5149	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
5	1, 3, 4	elab2 3626	1 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ⟨cop 4574  {copab 5148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-opab 5149

This theorem is referenced by:  rexopabb  5477  ssopab2bw  5496  dmopab  5865  rnopab  5904  funopab  6528  opabiota  6917  fvopab5  6976  f1ompt  7058  ovid  7502  zfrep6OLD  7902  enssdomOLD  8918  omxpenlem  9010  infxpenlem  9929  canthwelem  10567  pospo  18303  2ndcdisj  23434  lgsquadlem1  27360  lgsquadlem2  27361  h2hlm  31069  opabdm  32702  opabrn  32703  fpwrelmap  32824  eulerpartlemgvv  34539  fineqvrep  35277  satfvsucsuc  35566  bj-opelopabid  37520  phpreu  37942  poimirlem26  37984  vvdifopab  38603  brabidgaw  38711  diclspsn  41657  areaquad  43665  sprsymrelf  47970