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Theorem elqsg 6531
Description: Closed form of elqs 6532. (Contributed by Rodolfo Medina, 12-Oct-2010.)
Assertion
Ref Expression
elqsg  |-  ( B  e.  V  ->  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R
) )
Distinct variable groups:    x, A    x, B    x, R
Allowed substitution hint:    V( x)

Proof of Theorem elqsg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2164 . . 3  |-  ( y  =  B  ->  (
y  =  [ x ] R  <->  B  =  [
x ] R ) )
21rexbidv 2458 . 2  |-  ( y  =  B  ->  ( E. x  e.  A  y  =  [ x ] R  <->  E. x  e.  A  B  =  [ x ] R ) )
3 df-qs 6487 . 2  |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
42, 3elab2g 2859 1  |-  ( B  e.  V  ->  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1335    e. wcel 2128   E.wrex 2436   [cec 6479   /.cqs 6480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-qs 6487
This theorem is referenced by:  elqs  6532  elqsi  6533  ecelqsg  6534
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