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Theorem dfrab2 3275
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 2369 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 inab 3268 . . 3  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  { x  |  ( x  e.  A  /\  ph ) }
3 abid2 2209 . . . 4  |-  { x  |  x  e.  A }  =  A
43ineq1i 3198 . . 3  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  ( A  i^i  { x  | 
ph } )
52, 4eqtr3i 2111 . 2  |-  { x  |  ( x  e.  A  /\  ph ) }  =  ( A  i^i  { x  |  ph } )
6 incom 3193 . 2  |-  ( A  i^i  { x  | 
ph } )  =  ( { x  | 
ph }  i^i  A
)
71, 5, 63eqtri 2113 1  |-  { x  e.  A  |  ph }  =  ( { x  |  ph }  i^i  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1290    e. wcel 1439   {cab 2075   {crab 2364    i^i cin 2999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rab 2369  df-v 2622  df-in 3006
This theorem is referenced by:  minmax  10715
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