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Theorem sbcel21v 3039
Description: Class substitution into a membership relation. One direction of sbcel2gv 3038 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 2983 . 2  |-  ( [. B  /  x ]. A  e.  x  ->  B  e. 
_V )
2 sbcel2gv 3038 . . 3  |-  ( B  e.  _V  ->  ( [. B  /  x ]. A  e.  x  <->  A  e.  B ) )
32biimpd 144 . 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 2158   _Vcvv 2749   [.wsbc 2974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-sbc 2975
This theorem is referenced by: (None)
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