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Theorem sbcel2gv 3315
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 2538 . 2  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
2 eleq2 2538 . 2  |-  ( y  =  B  ->  ( A  e.  y  <->  A  e.  B ) )
31, 2sbcie2g 3289 1  |-  ( B  e.  V  ->  ( [. B  /  x ]. A  e.  x  <->  A  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    e. wcel 1904   [.wsbc 3255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-v 3033  df-sbc 3256
This theorem is referenced by:  sbcel21v  3316  csbvarg  3796  bnj92  29745  bnj539  29774  frege77  36607  sbcoreleleq  36966  trsbc  36971  onfrALTlem5  36978  sbcoreleleqVD  37319  trsbcVD  37337  onfrALTlem5VD  37345
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