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Theorem rspcsbela 3054
Description: Special case related to rspsbc 2986. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
Assertion
Ref Expression
rspcsbela  |-  ( ( A  e.  B  /\  A. x  e.  B  C  e.  D )  ->  [_ A  /  x ]_ C  e.  D )
Distinct variable groups:    x, B    x, D
Allowed substitution hints:    A( x)    C( x)

Proof of Theorem rspcsbela
StepHypRef Expression
1 rspsbc 2986 . . 3  |-  ( A  e.  B  ->  ( A. x  e.  B  C  e.  D  ->  [. A  /  x ]. C  e.  D )
)
2 sbcel1g 3016 . . 3  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  e.  D  <->  [_ A  /  x ]_ C  e.  D )
)
31, 2sylibd 148 . 2  |-  ( A  e.  B  ->  ( A. x  e.  B  C  e.  D  ->  [_ A  /  x ]_ C  e.  D )
)
43imp 123 1  |-  ( ( A  e.  B  /\  A. x  e.  B  C  e.  D )  ->  [_ A  /  x ]_ C  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1480   A.wral 2414   [.wsbc 2904   [_csb 2998
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-sbc 2905  df-csb 2999
This theorem is referenced by:  fsumzcl2  11167  fsumsplitsnun  11181  modfsummodlem1  11218  mulcncflem  12748  mulcncf  12749
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