MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  inab Structured version   Unicode version

Theorem inab 3611
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 2141 . . 3  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
2 df-clab 2425 . . 3  |-  ( y  e.  { x  |  ( ph  /\  ps ) }  <->  [ y  /  x ] ( ph  /\  ps ) )
3 df-clab 2425 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
4 df-clab 2425 . . . 4  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
53, 4anbi12i 680 . . 3  |-  ( ( y  e.  { x  |  ph }  /\  y  e.  { x  |  ps } )  <->  ( [
y  /  x ] ph  /\  [ y  /  x ] ps ) )
61, 2, 53bitr4ri 271 . 2  |-  ( ( y  e.  { x  |  ph }  /\  y  e.  { x  |  ps } )  <->  y  e.  { x  |  ( ph  /\ 
ps ) } )
76ineqri 3536 1  |-  ( { x  |  ph }  i^i  { x  |  ps } )  =  {
x  |  ( ph  /\ 
ps ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653   [wsb 1659    e. wcel 1726   {cab 2424    i^i cin 3321
This theorem is referenced by:  inrab  3615  inrab2  3616  dfrab2  3618  dfrab3  3619  orduniss2  4815  ssenen  7283  hashf1lem2  11707  ballotlem2  24748  dfiota3  25770  diophin  26833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329
  Copyright terms: Public domain W3C validator