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Theorem rspcsbela 2987
Description: Special case related to rspsbc 2921. (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 2921 . . 3  |-  ( A  e.  B  ->  ( A. x  e.  B  C  e.  D  ->  [. A  /  x ]. C  e.  D )
)
2 sbcel1g 2950 . . 3  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  e.  D  <->  [_ A  /  x ]_ C  e.  D )
)
31, 2sylibd 147 . 2  |-  ( A  e.  B  ->  ( A. x  e.  B  C  e.  D  ->  [_ A  /  x ]_ C  e.  D )
)
43imp 122 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 102    e. wcel 1438   A.wral 2359   [.wsbc 2840   [_csb 2933
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-sbc 2841  df-csb 2934
This theorem is referenced by:  fsumzcl2  10799  fsumsplitsnun  10813  modfsummodlem1  10850
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