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Theorem rabn0r 3272
Description: Non-empty 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 3271 . 2  |-  ( E. x ( x  e.  A  /\  ph )  ->  { x  |  ( x  e.  A  /\  ph ) }  =/=  (/) )
2 df-rex 2329 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
3 df-rab 2332 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
43neeq1i 2235 . 2  |-  ( { x  e.  A  |  ph }  =/=  (/)  <->  { x  |  ( x  e.  A  /\  ph ) }  =/=  (/) )
51, 2, 43imtr4i 194 1  |-  ( E. x  e.  A  ph  ->  { x  e.  A  |  ph }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   E.wex 1397    e. wcel 1409   {cab 2042    =/= wne 2220   E.wrex 2324   {crab 2327   (/)c0 3252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-nul 3253
This theorem is referenced by: (None)
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