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Theorem inab 3436
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 2009 . . 3  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
2 df-clab 2270 . . 3  |-  ( y  e.  { x  |  ( ph  /\  ps ) }  <->  [ y  /  x ] ( ph  /\  ps ) )
3 df-clab 2270 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
4 df-clab 2270 . . . 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 3362 1  |-  ( { x  |  ph }  i^i  { x  |  ps } )  =  {
x  |  ( ph  /\ 
ps ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623   [wsb 1629    e. wcel 1684   {cab 2269    i^i cin 3151
This theorem is referenced by:  inrab  3440  inrab2  3441  dfrab2  3443  dfrab3  3444  orduniss2  4624  ssenen  7035  hashf1lem2  11394  ballotlem2  23047  dfiota3  24462  mapex2  25140  diophin  26852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159
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