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Theorem sbcel1gv 3050
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 2994 . 2  |-  ( y  =  A  ->  ( [ y  /  x ] x  e.  B  <->  [. A  /  x ]. x  e.  B )
)
2 eleq1 2343 . 2  |-  ( y  =  A  ->  (
y  e.  B  <->  A  e.  B ) )
3 clelsb3 2385 . 2  |-  ( [ y  /  x ]
x  e.  B  <->  y  e.  B )
41, 2, 3vtoclbg 2844 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 1629    e. wcel 1684   [.wsbc 2991
This theorem is referenced by:  tfinds2  4654  filuni  17580  iuninc  23158  sbcoreleleq  28298  onfrALTlem4  28308  sbcoreleleqVD  28635  onfrALTlem4VD  28662  bnj110  28890
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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