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Theorem sbcex 3000
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 2992 . 2  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
2 elex 2796 . 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 1684   {cab 2269   _Vcvv 2788   [.wsbc 2991
This theorem is referenced by:  sbcco  3013  sbc5  3015  sbcan  3033  sbcor  3035  sbcal  3038  sbcex2  3040  spesbc  3072  opelopabsb  4275  sbccomieg  26870
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790  df-sbc 2992
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