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Theorem sbcel2gv 3213
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 2496 . 2  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
2 eleq2 2496 . 2  |-  ( y  =  B  ->  ( A  e.  y  <->  A  e.  B ) )
31, 2sbcie2g 3186 1  |-  ( B  e.  V  ->  ( [. B  /  x ]. A  e.  x  <->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1725   [.wsbc 3153
This theorem is referenced by:  csbvarg  3270  sbcoreleleq  28546  trsbc  28552  onfrALTlem5  28555  sbcoreleleqVD  28898  trsbcVD  28916  onfrALTlem5VD  28924  bnj92  29160  bnj539  29189
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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