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Theorem dfrab3 3610
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 2707 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 inab 3602 . 2  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  { x  |  ( x  e.  A  /\  ph ) }
3 abid2 2553 . . 3  |-  { x  |  x  e.  A }  =  A
43ineq1i 3531 . 2  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  ( A  i^i  { x  | 
ph } )
51, 2, 43eqtr2i 2462 1  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   {crab 2702    i^i cin 3312
This theorem is referenced by:  notrab  3611  dfrab3ss  3612  dfif3  3742  dffr3  5229  dfse2  5230  rabfi  7326  dfsup2  7440  ressmplbas2  16511  clsocv  19197  hasheuni  24468  tz6.26  25473  uvcff  27209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2707  df-v 2951  df-in 3320
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