ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbcel21v Unicode version

Theorem sbcel21v 3042
Description: Class substitution into a membership relation. One direction of sbcel2gv 3041 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 2986 . 2  |-  ( [. B  /  x ]. A  e.  x  ->  B  e. 
_V )
2 sbcel2gv 3041 . . 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 2160   _Vcvv 2752   [.wsbc 2977
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator