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Theorem rspcsbela 3153
Description: Special case related to rspsbc 3082. (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 3082 . . 3  |-  ( A  e.  B  ->  ( A. x  e.  B  C  e.  D  ->  [. A  /  x ]. C  e.  D )
)
2 sbcel1g 3113 . . 3  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  e.  D  <->  [_ A  /  x ]_ C  e.  D )
)
31, 2sylibd 205 . 2  |-  ( A  e.  B  ->  ( A. x  e.  B  C  e.  D  ->  [_ A  /  x ]_ C  e.  D )
)
43imp 418 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 358    e. wcel 1696   A.wral 2556   [.wsbc 3004   [_csb 3094
This theorem is referenced by:  iunmbl2  18930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-sbc 3005  df-csb 3095
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