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Theorem sbcel1gv 3212
Description: Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcel1gv  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  e.  B  <->  A  e.  B ) )
Distinct variable group:    x, B
Allowed substitution hints:    A( x)    V( x)

Proof of Theorem sbcel1gv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3156 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] x  e.  B  <->  [. A  /  x ]. x  e.  B )
)
2 eleq1 2495 . 2  |-  ( y  =  A  ->  (
y  e.  B  <->  A  e.  B ) )
3 clelsb3 2537 . 2  |-  ( [ y  /  x ]
x  e.  B  <->  y  e.  B )
41, 2, 3vtoclbg 3004 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  e.  B  <->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   [wsb 1658    e. wcel 1725   [.wsbc 3153
This theorem is referenced by:  tfinds2  4835  filuni  17909  sbcoreleleq  28556  onfrALTlem4  28566  sbcoreleleqVD  28908  onfrALTlem4VD  28935  bnj110  29166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154
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