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Theorem opabid 4453
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 4419 . 2  |-  <. x ,  y >.  e.  _V
2 copsexg 4434 . . 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 4259 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
51, 3, 4elab2 3077 1  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   <.cop 3809   {copab 4257
This theorem is referenced by:  opelopabsb  4457  ssopab2b  4473  dmopab  5072  rnopab  5107  funopab  5478  f1ompt  5883  zfrep6  5960  ovid  6182  fvopab5  6526  opabiota  6530  enssdom  7124  omxpenlem  7201  infxpenlem  7887  canthwelem  8517  pospo  14422  2ndcdisj  17511  lgsquadlem1  21130  lgsquadlem2  21131  h2hlm  22475  diclspsn  31929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259
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