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Theorem opabid 3968
Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
opabid (⟨x, y {⟨x, y⟩ ∣ φ} ↔ φ)

Proof of Theorem opabid
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 vex 2538 . . 3 x V
2 vex 2538 . . 3 y V
31, 2opex 3940 . 2 x, y V
4 copsexg 3955 . . 3 (z = ⟨x, y⟩ → (φxy(z = ⟨x, y φ)))
54bicomd 129 . 2 (z = ⟨x, y⟩ → (xy(z = ⟨x, y φ) ↔ φ))
6 df-opab 3793 . 2 {⟨x, y⟩ ∣ φ} = {zxy(z = ⟨x, y φ)}
73, 5, 6elab2 2667 1 (⟨x, y {⟨x, y⟩ ∣ φ} ↔ φ)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374  cop 3353  {copab 3791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-opab 3793
This theorem is referenced by:  opelopabsb  3971  ssopab2b  3987  dmopab  4473  rnopab  4508  funopab  4861  funco  4866  fvmptss2  5172  f1ompt  5245  ovid  5540
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