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Theorem elabd 2829
Description: Explicit demonstration the class  { x  |  ps } is not empty by the example  X. (Contributed by RP, 12-Aug-2020.)
Hypotheses
Ref Expression
elab.xex  |-  ( ph  ->  X  e.  _V )
elab.xmaj  |-  ( ph  ->  ch )
elab.xsub  |-  ( x  =  X  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
elabd  |-  ( ph  ->  E. x ps )
Distinct variable groups:    ch, x    x, X
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem elabd
StepHypRef Expression
1 elab.xex . 2  |-  ( ph  ->  X  e.  _V )
2 elab.xmaj . 2  |-  ( ph  ->  ch )
3 elab.xsub . . 3  |-  ( x  =  X  ->  ( ps 
<->  ch ) )
43spcegv 2774 . 2  |-  ( X  e.  _V  ->  ( ch  ->  E. x ps )
)
51, 2, 4sylc 62 1  |-  ( ph  ->  E. x ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331   E.wex 1468    e. wcel 1480   _Vcvv 2686
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688
This theorem is referenced by:  ntrivcvgap0  11330
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