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Theorem sbcex 3013
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 3005 . 2  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
2 elex 2809 . 2  |-  ( A  e.  { x  | 
ph }  ->  A  e.  _V )
31, 2sylbi 187 1  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   {cab 2282   _Vcvv 2801   [.wsbc 3004
This theorem is referenced by:  sbcco  3026  sbc5  3028  sbcan  3046  sbcor  3048  sbcal  3051  sbcex2  3053  spesbc  3085  opelopabsb  4291  sbccomieg  26973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803  df-sbc 3005
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