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Theorem sbcel1v 3383
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 3334 . 2  |-  ( [. A  /  x ]. x  e.  B  ->  A  e. 
_V )
2 elex 3115 . 2  |-  ( A  e.  B  ->  A  e.  _V )
3 dfsbcq2 3327 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] x  e.  B  <->  [. A  /  x ]. x  e.  B )
)
4 eleq1 2526 . . 3  |-  ( y  =  A  ->  (
y  e.  B  <->  A  e.  B ) )
5 clelsb3 2575 . . 3  |-  ( [ y  /  x ]
x  e.  B  <->  y  e.  B )
63, 4, 5vtoclbg 3165 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. x  e.  B  <->  A  e.  B ) )
71, 2, 6pm5.21nii 351 1  |-  ( [. A  /  x ]. x  e.  B  <->  A  e.  B
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   [wsb 1744    e. wcel 1823   _Vcvv 3106   [.wsbc 3324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3108  df-sbc 3325
This theorem is referenced by:  tfinds2  6671  filuni  20552  f1od2  27778  esum2dlem  28321  sbcoreleleq  33696  onfrALTlem4  33709  bnj110  34317  cotrclrcl  38232
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