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Theorem opabid 4421
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 >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )

Proof of Theorem opabid
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 opex 4387 . 2  |-  <. x ,  y >.  e.  _V
2 copsexg 4402 . . 3  |-  ( z  =  <. x ,  y
>.  ->  ( ph  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) ) )
32bicomd 193 . 2  |-  ( z  =  <. x ,  y
>.  ->  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ph )  <->  ph ) )
4 df-opab 4227 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
51, 3, 4elab2 3045 1  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   <.cop 3777   {copab 4225
This theorem is referenced by:  opelopabsb  4425  ssopab2b  4441  dmopab  5039  rnopab  5074  funopab  5445  f1ompt  5850  zfrep6  5927  ovid  6149  fvopab5  6493  opabiota  6497  enssdom  7091  omxpenlem  7168  infxpenlem  7851  canthwelem  8481  pospo  14385  2ndcdisj  17472  lgsquadlem1  21091  lgsquadlem2  21092  h2hlm  22436  diclspsn  31677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-opab 4227
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