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Theorem sbcel1gv 3222
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 3166 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] x  e.  B  <->  [. A  /  x ]. x  e.  B )
)
2 eleq1 2498 . 2  |-  ( y  =  A  ->  (
y  e.  B  <->  A  e.  B ) )
3 clelsb3 2540 . 2  |-  ( [ y  /  x ]
x  e.  B  <->  y  e.  B )
41, 2, 3vtoclbg 3014 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  e.  B  <->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   [wsb 1659    e. wcel 1726   [.wsbc 3163
This theorem is referenced by:  tfinds2  4846  filuni  17922  sbcoreleleq  28693  onfrALTlem4  28703  sbcoreleleqVD  29045  onfrALTlem4VD  29072  bnj110  29303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sbc 3164
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