MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcel2gv Unicode version

Theorem sbcel2gv 3164
Description: Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcel2gv  |-  ( B  e.  V  ->  ( [. B  /  x ]. A  e.  x  <->  A  e.  B ) )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem sbcel2gv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eleq2 2448 . 2  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
2 eleq2 2448 . 2  |-  ( y  =  B  ->  ( A  e.  y  <->  A  e.  B ) )
31, 2sbcie2g 3137 1  |-  ( B  e.  V  ->  ( [. B  /  x ]. A  e.  x  <->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1717   [.wsbc 3104
This theorem is referenced by:  csbvarg  3221  sbcoreleleq  27962  trsbc  27968  onfrALTlem5  27971  sbcoreleleqVD  28312  trsbcVD  28330  onfrALTlem5VD  28338  bnj92  28571  bnj539  28600
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-sbc 3105
  Copyright terms: Public domain W3C validator