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Theorem inrab2 3317
Description: Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
Assertion
Ref Expression
inrab2  |-  ( { x  e.  A  |  ph }  i^i  B )  =  { x  e.  ( A  i^i  B
)  |  ph }
Distinct variable group:    x, B
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem inrab2
StepHypRef Expression
1 df-rab 2400 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 abid2 2236 . . . 4  |-  { x  |  x  e.  B }  =  B
32eqcomi 2119 . . 3  |-  B  =  { x  |  x  e.  B }
41, 3ineq12i 3243 . 2  |-  ( { x  e.  A  |  ph }  i^i  B )  =  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B }
)
5 df-rab 2400 . . 3  |-  { x  e.  ( A  i^i  B
)  |  ph }  =  { x  |  ( x  e.  ( A  i^i  B )  /\  ph ) }
6 inab 3312 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B } )  =  {
x  |  ( ( x  e.  A  /\  ph )  /\  x  e.  B ) }
7 elin 3227 . . . . . . 7  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
87anbi1i 451 . . . . . 6  |-  ( ( x  e.  ( A  i^i  B )  /\  ph )  <->  ( ( x  e.  A  /\  x  e.  B )  /\  ph ) )
9 an32 534 . . . . . 6  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  ph )  <->  ( (
x  e.  A  /\  ph )  /\  x  e.  B ) )
108, 9bitri 183 . . . . 5  |-  ( ( x  e.  ( A  i^i  B )  /\  ph )  <->  ( ( x  e.  A  /\  ph )  /\  x  e.  B
) )
1110abbii 2231 . . . 4  |-  { x  |  ( x  e.  ( A  i^i  B
)  /\  ph ) }  =  { x  |  ( ( x  e.  A  /\  ph )  /\  x  e.  B
) }
126, 11eqtr4i 2139 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B } )  =  {
x  |  ( x  e.  ( A  i^i  B )  /\  ph ) }
135, 12eqtr4i 2139 . 2  |-  { x  e.  ( A  i^i  B
)  |  ph }  =  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B }
)
144, 13eqtr4i 2139 1  |-  ( { x  e.  A  |  ph }  i^i  B )  =  { x  e.  ( A  i^i  B
)  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1314    e. wcel 1463   {cab 2101   {crab 2395    i^i cin 3038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400  df-v 2660  df-in 3045
This theorem is referenced by:  iooval2  9638  fzval2  9733  dfphi2  11791  znnen  11806
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