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Theorem sbcel2gv 3336
Description: Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcel2gv  |-  ( B  e.  V  ->  ( [. B  /  x ]. A  e.  x  <->  A  e.  B ) )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem sbcel2gv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eleq2 2475 . 2  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
2 eleq2 2475 . 2  |-  ( y  =  B  ->  ( A  e.  y  <->  A  e.  B ) )
31, 2sbcie2g 3310 1  |-  ( B  e.  V  ->  ( [. B  /  x ]. A  e.  x  <->  A  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1842   [.wsbc 3276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-v 3060  df-sbc 3277
This theorem is referenced by:  sbcel21v  3337  csbvarg  3796  bnj92  29234  bnj539  29263  frege77  35901  sbcoreleleq  36306  trsbc  36311  onfrALTlem5  36318  sbcoreleleqVD  36670  trsbcVD  36688  onfrALTlem5VD  36696
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