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Theorem rspcsbela 3309
Description: Special case related to rspsbc 3240. (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 3240 . . 3  |-  ( A  e.  B  ->  ( A. x  e.  B  C  e.  D  ->  [. A  /  x ]. C  e.  D )
)
2 sbcel1g 3271 . . 3  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  e.  D  <->  [_ A  /  x ]_ C  e.  D )
)
31, 2sylibd 207 . 2  |-  ( A  e.  B  ->  ( A. x  e.  B  C  e.  D  ->  [_ A  /  x ]_ C  e.  D )
)
43imp 420 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 360    e. wcel 1726   A.wral 2706   [.wsbc 3162   [_csb 3252
This theorem is referenced by:  iunmbl2  19452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-v 2959  df-sbc 3163  df-csb 3253
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