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Theorem inrab2 3400
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 2457 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 abid2 2291 . . . 4  |-  { x  |  x  e.  B }  =  B
32eqcomi 2174 . . 3  |-  B  =  { x  |  x  e.  B }
41, 3ineq12i 3326 . 2  |-  ( { x  e.  A  |  ph }  i^i  B )  =  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B }
)
5 df-rab 2457 . . 3  |-  { x  e.  ( A  i^i  B
)  |  ph }  =  { x  |  ( x  e.  ( A  i^i  B )  /\  ph ) }
6 inab 3395 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B } )  =  {
x  |  ( ( x  e.  A  /\  ph )  /\  x  e.  B ) }
7 elin 3310 . . . . . . 7  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
87anbi1i 455 . . . . . 6  |-  ( ( x  e.  ( A  i^i  B )  /\  ph )  <->  ( ( x  e.  A  /\  x  e.  B )  /\  ph ) )
9 an32 557 . . . . . 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 2286 . . . 4  |-  { x  |  ( x  e.  ( A  i^i  B
)  /\  ph ) }  =  { x  |  ( ( x  e.  A  /\  ph )  /\  x  e.  B
) }
126, 11eqtr4i 2194 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B } )  =  {
x  |  ( x  e.  ( A  i^i  B )  /\  ph ) }
135, 12eqtr4i 2194 . 2  |-  { x  e.  ( A  i^i  B
)  |  ph }  =  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B }
)
144, 13eqtr4i 2194 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 1348    e. wcel 2141   {cab 2156   {crab 2452    i^i cin 3120
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-in 3127
This theorem is referenced by:  iooval2  9859  fzval2  9955  dfphi2  12161  znnen  12340
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