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Theorem rabn0r 3359
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 3357 . 2  |-  ( E. x ( x  e.  A  /\  ph )  ->  { x  |  ( x  e.  A  /\  ph ) }  =/=  (/) )
2 df-rex 2399 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
3 df-rab 2402 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43neeq1i 2300 . 2  |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  { x  |  ( x  e.  A  /\  ph ) }  =/=  (/) )
51, 2, 43imtr4i 200 1  |-  ( E. x  e.  A  ph  ->  { x  e.  A  |  ph }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1453    e. wcel 1465   {cab 2103    =/= wne 2285   E.wrex 2394   {crab 2397   (/)c0 3333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-nul 3334
This theorem is referenced by: (None)
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