Theorem opabid 4393

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		vex 2824	. . 3 |- x e. _V
2		vex 2824	. . 3 |- y e. _V
3	1, 2	opex 4364	. 2 |- <. x , y >. e. _V
4		copsexg 4379	. . 3 |- ( z = <. x , y >. -> ( ph <-> E. x E. y ( z = <. x , y >. /\ ph ) ) )
5	4	bicomd 141	. 2 |- ( z = <. x , y >. -> ( E. x E. y ( z = <. x , y >. /\ ph ) <-> ph ) )
6		df-opab 4188	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
7	3, 5, 6	elab2 2974	1 |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   <.cop 3708   {copab 4186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-opab 4188

This theorem is referenced by:  opabidw  4394  opelopabsb  4397  ssopab2b  4414  dmopab  4987  rnopab  5024  funopab  5407  funco  5412  fvmptss2  5774  f1ompt  5850  ovid  6195  enssdom  7038