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Theorem dfrab3 3457
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 2565 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 inab 3449 . 2  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  { x  |  ( x  e.  A  /\  ph ) }
3 abid2 2413 . . 3  |-  { x  |  x  e.  A }  =  A
43ineq1i 3379 . 2  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  ( A  i^i  { x  | 
ph } )
51, 2, 43eqtr2i 2322 1  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560    i^i cin 3164
This theorem is referenced by:  notrab  3458  dfrab3ss  3459  dfif3  3588  dffr3  5061  dfse2  5062  dfsup2  7211  ressmplbas2  16215  clsocv  18693  rabfi  23186  hasheuni  23468  tz6.26  24276  dfdir2  25394  uvcff  27343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-in 3172
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