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Theorem inab 3449
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 2022 . . 3  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
2 df-clab 2283 . . 3  |-  ( y  e.  { x  |  ( ph  /\  ps ) }  <->  [ y  /  x ] ( ph  /\  ps ) )
3 df-clab 2283 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
4 df-clab 2283 . . . 4  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
53, 4anbi12i 678 . . 3  |-  ( ( y  e.  { x  |  ph }  /\  y  e.  { x  |  ps } )  <->  ( [
y  /  x ] ph  /\  [ y  /  x ] ps ) )
61, 2, 53bitr4ri 269 . 2  |-  ( ( y  e.  { x  |  ph }  /\  y  e.  { x  |  ps } )  <->  y  e.  { x  |  ( ph  /\ 
ps ) } )
76ineqri 3375 1  |-  ( { x  |  ph }  i^i  { x  |  ps } )  =  {
x  |  ( ph  /\ 
ps ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632   [wsb 1638    e. wcel 1696   {cab 2282    i^i cin 3164
This theorem is referenced by:  inrab  3453  inrab2  3454  dfrab2  3456  dfrab3  3457  orduniss2  4640  ssenen  7051  hashf1lem2  11410  ballotlem2  23063  dfiota3  24533  mapex2  25243  diophin  26955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172
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