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Theorem rabn0r 3307
Description: Nonempty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
Assertion
Ref Expression
rabn0r  |-  ( E. x  e.  A  ph  ->  { x  e.  A  |  ph }  =/=  (/) )

Proof of Theorem rabn0r
StepHypRef Expression
1 abn0r 3305 . 2  |-  ( E. x ( x  e.  A  /\  ph )  ->  { x  |  ( x  e.  A  /\  ph ) }  =/=  (/) )
2 df-rex 2365 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
3 df-rab 2368 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43neeq1i 2270 . 2  |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  { x  |  ( x  e.  A  /\  ph ) }  =/=  (/) )
51, 2, 43imtr4i 199 1  |-  ( E. x  e.  A  ph  ->  { x  e.  A  |  ph }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   E.wex 1426    e. wcel 1438   {cab 2074    =/= wne 2255   E.wrex 2360   {crab 2363   (/)c0 3284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-nul 3285
This theorem is referenced by: (None)
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