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Theorem opabid 4270
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 4236 . 2  |-  <. x ,  y >.  e.  _V
2 copsexg 4251 . . 3  |-  ( z  =  <. x ,  y
>.  ->  ( ph  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) ) )
32bicomd 192 . 2  |-  ( z  =  <. x ,  y
>.  ->  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ph )  <->  ph ) )
4 df-opab 4079 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
51, 3, 4elab2 2918 1  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1685   <.cop 3644   {copab 4077
This theorem is referenced by:  opelopabsb  4274  ssopab2b  4290  dmopab  4888  rnopab  4923  funopab  5253  f1ompt  5644  zfrep6  5710  ovid  5926  fvopab5  6283  opabiota  6287  enssdom  6882  omxpenlem  6959  infxpenlem  7637  canthwelem  8268  pospo  14103  2ndcdisj  17178  lgsquadlem1  20589  lgsquadlem2  20590  h2hlm  21556  bosser  25578  diclspsn  30663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-opab 4079
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