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Theorem inab 3250
Description: Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
inab  |-  ( { x  |  ph }  i^i  { x  |  ps } )  =  {
x  |  ( ph  /\ 
ps ) }

Proof of Theorem inab
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sban 1872 . . 3  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
2 df-clab 2070 . . 3  |-  ( y  e.  { x  |  ( ph  /\  ps ) }  <->  [ y  /  x ] ( ph  /\  ps ) )
3 df-clab 2070 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
4 df-clab 2070 . . . 4  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
53, 4anbi12i 448 . . 3  |-  ( ( y  e.  { x  |  ph }  /\  y  e.  { x  |  ps } )  <->  ( [
y  /  x ] ph  /\  [ y  /  x ] ps ) )
61, 2, 53bitr4ri 211 . 2  |-  ( ( y  e.  { x  |  ph }  /\  y  e.  { x  |  ps } )  <->  y  e.  { x  |  ( ph  /\ 
ps ) } )
76ineqri 3177 1  |-  ( { x  |  ph }  i^i  { x  |  ps } )  =  {
x  |  ( ph  /\ 
ps ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285    e. wcel 1434   [wsb 1687   {cab 2069    i^i cin 2983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-in 2990
This theorem is referenced by:  inrab  3254  inrab2  3255  dfrab2  3257  dfrab3  3258  imainlem  5046  imain  5047  ssenen  6495
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