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Theorem inrab 3765
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 2818 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-rab 2818 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
31, 2ineq12i 3693 . 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 2818 . . 3  |-  { x  e.  A  |  ( ph  /\  ps ) }  =  { x  |  ( x  e.  A  /\  ( ph  /\  ps ) ) }
5 inab 3761 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  /\  ( x  e.  A  /\  ps ) ) }
6 anandi 825 . . . . 5  |-  ( ( x  e.  A  /\  ( ph  /\  ps )
)  <->  ( ( x  e.  A  /\  ph )  /\  ( x  e.  A  /\  ps )
) )
76abbii 2596 . . . 4  |-  { x  |  ( x  e.  A  /\  ( ph  /\ 
ps ) ) }  =  { x  |  ( ( x  e.  A  /\  ph )  /\  ( x  e.  A  /\  ps ) ) }
85, 7eqtr4i 2494 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( x  e.  A  /\  ( ph  /\  ps ) ) }
94, 8eqtr4i 2494 . 2  |-  { x  e.  A  |  ( ph  /\  ps ) }  =  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  ( x  e.  A  /\  ps ) } )
103, 9eqtr4i 2494 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 369    = wceq 1374    e. wcel 1762   {cab 2447   {crab 2813    i^i cin 3470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-rab 2818  df-v 3110  df-in 3478
This theorem is referenced by:  rabnc  3804  ixxin  11537  hashbclem  12456  phiprmpw  14156  submacs  15801  ablfacrp  16902  dfrhm2  17145  ordtbaslem  19450  ordtbas2  19453  ordtopn3  19458  ordtcld3  19461  ordthauslem  19645  pthaus  19869  xkohaus  19884  tsmsfbas  20356  minveclem3b  21573  shftmbl  21679  mumul  23178  ppiub  23202  lgsquadlem2  23353  cusgrasizeindslem2  24138  frisusgranb  24661  numclwwlkdisj  24745  numclwwlk3lem  24773  xppreima  27147  xpinpreima  27512  xpinpreima2  27513  measvuni  27813  subfacp1lem6  28257  cnambfre  29629  itg2addnclem2  29633  ftc1anclem6  29661  anrabdioph  30307
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