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Theorem sbcex 2920
Description: By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbcex  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )

Proof of Theorem sbcex
StepHypRef Expression
1 df-sbc 2913 . 2  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
2 elex 2700 . 2  |-  ( A  e.  { x  | 
ph }  ->  A  e.  _V )
31, 2sylbi 120 1  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1481   {cab 2126   _Vcvv 2689   [.wsbc 2912
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-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691  df-sbc 2913
This theorem is referenced by:  sbcco  2933  sbc5  2935  sbcan  2954  sbcor  2956  sbcn1  2959  sbcim1  2960  sbcbi1  2961  sbcal  2963  sbcex2  2965  sbcel1v  2974  sbcel21v  2976  sbcimdv  2977  sbcrext  2989  spesbc  2997  csbprc  3412  opelopabsb  4189
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