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Theorem inab 3729
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 2101 . . 3  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
2 df-clab 2440 . . 3  |-  ( y  e.  { x  |  ( ph  /\  ps ) }  <->  [ y  /  x ] ( ph  /\  ps ) )
3 df-clab 2440 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
4 df-clab 2440 . . . 4  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
53, 4anbi12i 697 . . 3  |-  ( ( y  e.  { x  |  ph }  /\  y  e.  { x  |  ps } )  <->  ( [
y  /  x ] ph  /\  [ y  /  x ] ps ) )
61, 2, 53bitr4ri 278 . 2  |-  ( ( y  e.  { x  |  ph }  /\  y  e.  { x  |  ps } )  <->  y  e.  { x  |  ( ph  /\ 
ps ) } )
76ineqri 3655 1  |-  ( { x  |  ph }  i^i  { x  |  ps } )  =  {
x  |  ( ph  /\ 
ps ) }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370   [wsb 1702    e. wcel 1758   {cab 2439    i^i cin 3438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-in 3446
This theorem is referenced by:  inrab  3733  inrab2  3734  dfrab3  3736  orduniss2  6557  ssenen  7598  hashf1lem2  12330  ballotlem2  27035  dfiota3  28118  ptrest  28593  diophin  29279  bj-inrab  32782
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