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Theorem dfrab3 3213
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 2315 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 inab 3205 . 2  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  { x  |  ( x  e.  A  /\  ph ) }
3 abid2 2158 . . 3  |-  { x  |  x  e.  A }  =  A
43ineq1i 3134 . 2  |-  ( { x  |  x  e.  A }  i^i  {
x  |  ph }
)  =  ( A  i^i  { x  | 
ph } )
51, 2, 43eqtr2i 2066 1  |-  { x  e.  A  |  ph }  =  ( A  i^i  { x  |  ph }
)
Colors of variables: wff set class
Syntax hints:    /\ wa 97    = wceq 1243    e. wcel 1393   {cab 2026   {crab 2310    i^i cin 2916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-in 2924
This theorem is referenced by:  notrab  3214  dfrab3ss  3215  dfif3  3343  dfse2  4698
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