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Theorem inrab 3721
Description: Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
Assertion
Ref Expression
inrab  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  ps } )  =  {
x  e.  A  | 
( ph  /\  ps ) }

Proof of Theorem inrab
StepHypRef Expression
1 df-rab 2762 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-rab 2762 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
31, 2ineq12i 3638 . 2  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  ps } )  =  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  {
x  |  ( x  e.  A  /\  ps ) } )
4 df-rab 2762 . . 3  |-  { x  e.  A  |  ( ph  /\  ps ) }  =  { x  |  ( x  e.  A  /\  ( ph  /\  ps ) ) }
5 inab 3717 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  /\  ( x  e.  A  /\  ps ) ) }
6 anandi 829 . . . . 5  |-  ( ( x  e.  A  /\  ( ph  /\  ps )
)  <->  ( ( x  e.  A  /\  ph )  /\  ( x  e.  A  /\  ps )
) )
76abbii 2536 . . . 4  |-  { x  |  ( x  e.  A  /\  ( ph  /\ 
ps ) ) }  =  { x  |  ( ( x  e.  A  /\  ph )  /\  ( x  e.  A  /\  ps ) ) }
85, 7eqtr4i 2434 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( x  e.  A  /\  ( ph  /\  ps ) ) }
94, 8eqtr4i 2434 . 2  |-  { x  e.  A  |  ( ph  /\  ps ) }  =  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  ( x  e.  A  /\  ps ) } )
103, 9eqtr4i 2434 1  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  ps } )  =  {
x  e.  A  | 
( ph  /\  ps ) }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1405    e. wcel 1842   {cab 2387   {crab 2757    i^i cin 3412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rab 2762  df-v 3060  df-in 3420
This theorem is referenced by:  rabnc  3762  ixxin  11517  hashbclem  12457  phiprmpw  14407  submacs  16212  ablfacrp  17329  dfrhm2  17578  ordtbaslem  19874  ordtbas2  19877  ordtopn3  19882  ordtcld3  19885  ordthauslem  20069  pthaus  20323  xkohaus  20338  tsmsfbas  20810  minveclem3b  22027  shftmbl  22133  mumul  23728  ppiub  23752  lgsquadlem2  23903  cusgrasizeindslem2  24772  frisusgranb  25295  numclwwlkdisj  25378  numclwwlk3lem  25406  xppreima  27810  xpinpreima  28221  xpinpreima2  28222  measvuni  28542  subfacp1lem6  29363  cnambfre  31416  itg2addnclem2  31421  ftc1anclem6  31449  anrabdioph  35056  undisjrab  36015
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