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Theorem sbcel2gv 3051
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 dfsbcq2 2994 . 2  |-  ( y  =  B  ->  ( [ y  /  x ] A  e.  x  <->  [. B  /  x ]. A  e.  x )
)
2 eleq2 2344 . 2  |-  ( y  =  B  ->  ( A  e.  y  <->  A  e.  B ) )
3 nfv 1605 . . 3  |-  F/ x  A  e.  y
4 eleq2 2344 . . 3  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
53, 4sbie 1978 . 2  |-  ( [ y  /  x ] A  e.  x  <->  A  e.  y )
61, 2, 5vtoclbg 2844 1  |-  ( B  e.  V  ->  ( [. B  /  x ]. A  e.  x  <->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   [wsb 1629    e. wcel 1684   [.wsbc 2991
This theorem is referenced by:  csbvarg  3108  sbcoreleleq  28298  trsbc  28304  onfrALTlem5  28307  sbcoreleleqVD  28635  trsbcVD  28653  onfrALTlem5VD  28661  bnj92  28894  bnj539  28923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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