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Theorem sbcex 2837
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 2830 . 2  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
2 elex 2624 . 2  |-  ( A  e.  { x  | 
ph }  ->  A  e.  _V )
31, 2sylbi 119 1  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1436   {cab 2071   _Vcvv 2615   [.wsbc 2829
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-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-v 2617  df-sbc 2830
This theorem is referenced by:  sbcco  2850  sbc5  2852  sbcan  2870  sbcor  2872  sbcn1  2875  sbcim1  2876  sbcbi1  2877  sbcal  2879  sbcex2  2881  sbcel1v  2890  sbcel21v  2892  sbcimdv  2893  sbcrext  2905  spesbc  2913  csbprc  3316  opelopabsb  4063
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