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Theorem dfrab3 3444
Description: Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
dfrab3  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem dfrab3
StepHypRef Expression
1 df-rab 2552 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 inab 3436 . 2  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  { x  |  ( x  e.  A  /\  ph ) }
3 abid2 2400 . . 3  |-  { x  |  x  e.  A }  =  A
43ineq1i 3366 . 2  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  ( A  i^i  { x  | 
ph } )
51, 2, 43eqtr2i 2309 1  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547    i^i cin 3151
This theorem is referenced by:  notrab  3445  dfrab3ss  3446  dfif3  3575  dffr3  5045  dfse2  5046  dfsup2  7195  ressmplbas2  16199  clsocv  18677  rabfi  23170  hasheuni  23453  tz6.26  24205  dfdir2  25291  uvcff  27240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-in 3159
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