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Theorem dfrab2 3246
Description: Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
Assertion
Ref Expression
dfrab2  |-  { x  e.  A  |  ph }  =  ( { x  |  ph }  i^i  A
)
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem dfrab2
StepHypRef Expression
1 df-rab 2358 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 inab 3239 . . 3  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  { x  |  ( x  e.  A  /\  ph ) }
3 abid2 2200 . . . 4  |-  { x  |  x  e.  A }  =  A
43ineq1i 3170 . . 3  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  ( A  i^i  { x  | 
ph } )
52, 4eqtr3i 2104 . 2  |-  { x  |  ( x  e.  A  /\  ph ) }  =  ( A  i^i  { x  |  ph } )
6 incom 3165 . 2  |-  ( A  i^i  { x  | 
ph } )  =  ( { x  | 
ph }  i^i  A
)
71, 5, 63eqtri 2106 1  |-  { x  e.  A  |  ph }  =  ( { x  |  ph }  i^i  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285    e. wcel 1434   {cab 2068   {crab 2353    i^i cin 2973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rab 2358  df-v 2604  df-in 2980
This theorem is referenced by:  minmax  10250
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