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Theorem sbcel21v 2903
Description: Class substitution into a membership relation. One direction of sbcel2gv 2902 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbcel21v  |-  ( [. B  /  x ]. A  e.  x  ->  A  e.  B )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem sbcel21v
StepHypRef Expression
1 sbcex 2848 . 2  |-  ( [. B  /  x ]. A  e.  x  ->  B  e. 
_V )
2 sbcel2gv 2902 . . 3  |-  ( B  e.  _V  ->  ( [. B  /  x ]. A  e.  x  <->  A  e.  B ) )
32biimpd 142 . 2  |-  ( B  e.  _V  ->  ( [. B  /  x ]. A  e.  x  ->  A  e.  B ) )
41, 3mpcom 36 1  |-  ( [. B  /  x ]. A  e.  x  ->  A  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438   _Vcvv 2619   [.wsbc 2840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841
This theorem is referenced by: (None)
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