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Theorem opabid 4149
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 vex 2663 . . 3  |-  x  e. 
_V
2 vex 2663 . . 3  |-  y  e. 
_V
31, 2opex 4121 . 2  |-  <. x ,  y >.  e.  _V
4 copsexg 4136 . . 3  |-  ( z  =  <. x ,  y
>.  ->  ( ph  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) ) )
54bicomd 140 . 2  |-  ( z  =  <. x ,  y
>.  ->  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ph )  <->  ph ) )
6 df-opab 3960 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
73, 5, 6elab2 2805 1  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1316   E.wex 1453    e. wcel 1465   <.cop 3500   {copab 3958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960
This theorem is referenced by:  opelopabsb  4152  ssopab2b  4168  dmopab  4720  rnopab  4756  funopab  5128  funco  5133  fvmptss2  5464  f1ompt  5539  ovid  5855  enssdom  6624
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