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Theorem sbcel1v 3396
Description: Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbcel1v  |-  ( [. A  /  x ]. x  e.  B  <->  A  e.  B
)
Distinct variable group:    x, B
Allowed substitution hint:    A( x)

Proof of Theorem sbcel1v
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sbcex 3341 . 2  |-  ( [. A  /  x ]. x  e.  B  ->  A  e. 
_V )
2 elex 3122 . 2  |-  ( A  e.  B  ->  A  e.  _V )
3 dfsbcq2 3334 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] x  e.  B  <->  [. A  /  x ]. x  e.  B )
)
4 eleq1 2539 . . 3  |-  ( y  =  A  ->  (
y  e.  B  <->  A  e.  B ) )
5 clelsb3 2588 . . 3  |-  ( [ y  /  x ]
x  e.  B  <->  y  e.  B )
63, 4, 5vtoclbg 3172 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. x  e.  B  <->  A  e.  B ) )
71, 2, 6pm5.21nii 353 1  |-  ( [. A  /  x ]. x  e.  B  <->  A  e.  B
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   [wsb 1711    e. wcel 1767   _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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115  df-sbc 3332
This theorem is referenced by:  tfinds2  6682  filuni  20149  f1od2  27247  sbcoreleleq  32403  onfrALTlem4  32413  bnj110  33013
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