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Theorem opabid 3985
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 2554 . . 3 x V
2 vex 2554 . . 3 y V
31, 2opex 3957 . 2 x, y V
4 copsexg 3972 . . 3 (z = ⟨x, y⟩ → (φxy(z = ⟨x, y φ)))
54bicomd 129 . 2 (z = ⟨x, y⟩ → (xy(z = ⟨x, y φ) ↔ φ))
6 df-opab 3810 . 2 {⟨x, y⟩ ∣ φ} = {zxy(z = ⟨x, y φ)}
73, 5, 6elab2 2684 1 (⟨x, y {⟨x, y⟩ ∣ φ} ↔ φ)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  cop 3370  {copab 3808
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810
This theorem is referenced by:  opelopabsb  3988  ssopab2b  4004  dmopab  4489  rnopab  4524  funopab  4878  funco  4883  fvmptss2  5190  f1ompt  5263  ovid  5559  enssdom  6178
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