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Theorem sbcex 3138
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 3130 . 2  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
2 elex 2932 . 2  |-  ( A  e.  { x  | 
ph }  ->  A  e.  _V )
31, 2sylbi 188 1  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721   {cab 2398   _Vcvv 2924   [.wsbc 3129
This theorem is referenced by:  sbcco  3151  sbc5  3153  sbcan  3171  sbcor  3173  sbcal  3176  sbcex2  3178  spesbc  3210  opelopabsb  4433  sbccomieg  26747
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-11 1757  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-v 2926  df-sbc 3130
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