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Theorem sbcel1gv 3126
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 3070 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] x  e.  B  <->  [. A  /  x ]. x  e.  B )
)
2 eleq1 2418 . 2  |-  ( y  =  A  ->  (
y  e.  B  <->  A  e.  B ) )
3 clelsb3 2460 . 2  |-  ( [ y  /  x ]
x  e.  B  <->  y  e.  B )
41, 2, 3vtoclbg 2920 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. x  e.  B  <->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   [wsb 1648    e. wcel 1710   [.wsbc 3067
This theorem is referenced by:  tfinds2  4736  filuni  17682  iuninc  23210  sbcoreleleq  28043  onfrALTlem4  28053  sbcoreleleqVD  28397  onfrALTlem4VD  28424  bnj110  28652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866  df-sbc 3068
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