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

Theorem sbcel2gv 3398
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 2540 . 2  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
2 eleq2 2540 . 2  |-  ( y  =  B  ->  ( A  e.  y  <->  A  e.  B ) )
31, 2sbcie2g 3365 1  |-  ( B  e.  V  ->  ( [. B  /  x ]. A  e.  x  <->  A  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1767   [.wsbc 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115  df-sbc 3332
This theorem is referenced by:  sbcel21v  3399  csbvarg  3848  sbcoreleleq  32385  trsbc  32391  onfrALTlem5  32394  sbcoreleleqVD  32739  trsbcVD  32757  onfrALTlem5VD  32765  bnj92  32999  bnj539  33028
  Copyright terms: Public domain W3C validator