Theorem opabid 4754

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.

Step	Hyp	Ref	Expression
1		opex 4711	. 2 |- <. x , y >. e. _V
2		copsexg 4732	. . 3 |- ( z = <. x , y >. -> ( ph <-> E. x E. y ( z = <. x , y >. /\ ph ) ) )
3	2	bicomd 201	. 2 |- ( z = <. x , y >. -> ( E. x E. y ( z = <. x , y >. /\ ph ) <-> ph ) )
4		df-opab 4506	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
5	1, 3, 4	elab2 3253	1 |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   <.cop 4033   {copab 4504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506

This theorem is referenced by:  opelopabsb  4757  ssopab2b  4774  dmopab  5211  rnopab  5245  funopab  5619  opabiota  5928  fvopab5  5971  f1ompt  6041  ovid  6401  zfrep6  6749  enssdom  7537  omxpenlem  7615  infxpenlem  8387  canthwelem  9024  pospo  15456  2ndcdisj  19723  lgsquadlem1  23357  lgsquadlem2  23358  h2hlm  25573  opabdm  27137  opabrn  27138  fpwrelmap  27228  eulerpartlemgvv  27955  areaquad  30789  diclspsn  35991