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Theorem sbcex 3341
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 3332 . 2  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
2 elex 3122 . 2  |-  ( A  e.  { x  | 
ph }  ->  A  e.  _V )
31, 2sylbi 195 1  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   {cab 2452   _Vcvv 3113   [.wsbc 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-12 1803  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115  df-sbc 3332
This theorem is referenced by:  sbcco  3354  sbc5  3356  sbcan  3374  sbcor  3376  sbcn1  3379  sbcim1  3380  sbcbi1  3381  sbcal  3383  sbcex2  3385  sbcel1v  3396  sbcel21v  3399  sbcimdv  3400  sbcrext  3414  sbcreu  3418  spesbc  3424  csbprc  3821  sbcel12  3823  sbcne12  3827  sbcel2  3831  sbccsb2  3851  sbcbr123  4498  opelopabsb  4757  csbopab  4779  csbxp  5080  csbiota  5579  csbriota  6256  sbccomieg  30346  bj-csbprc  33566
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