ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbcel1v Unicode version

Theorem sbcel1v 2877
Description: Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbcel1v  |-  ( [. A  /  x ]. x  e.  B  <->  A  e.  B
)
Distinct variable group:    x, B
Allowed substitution hint:    A( x)

Proof of Theorem sbcel1v
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sbcex 2824 . 2  |-  ( [. A  /  x ]. x  e.  B  ->  A  e. 
_V )
2 elex 2611 . 2  |-  ( A  e.  B  ->  A  e.  _V )
3 dfsbcq2 2819 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] x  e.  B  <->  [. A  /  x ]. x  e.  B )
)
4 eleq1 2142 . . 3  |-  ( y  =  A  ->  (
y  e.  B  <->  A  e.  B ) )
5 clelsb3 2184 . . 3  |-  ( [ y  /  x ]
x  e.  B  <->  y  e.  B )
63, 4, 5vtoclbg 2660 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. x  e.  B  <->  A  e.  B ) )
71, 2, 6pm5.21nii 653 1  |-  ( [. A  /  x ]. x  e.  B  <->  A  e.  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1434   [wsb 1686   _Vcvv 2602   [.wsbc 2816
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817
This theorem is referenced by:  f1od2  5881
  Copyright terms: Public domain W3C validator