Theorem rspcsbela 3128

Description: Special case related to rspsbc 3057. (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

Step	Hyp	Ref	Expression
1		rspsbc 3057	. . 3 |- ( A e. B -> ( A. x e. B C e. D -> [. A / x ]. C e. D ) )
2		sbcel1g 3088	. . 3 |- ( A e. B -> ( [. A / x ]. C e. D <-> [_ A / x ]_ C e. D ) )
3	1, 2	sylibd 149	. 2 |- ( A e. B -> ( A. x e. B C e. D -> [_ A / x ]_ C e. D ) )
4	3	imp 124	1 |- ( ( A e. B /\ A. x e. B C e. D ) -> [_ A / x ]_ C e. D )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2158   A.wral 2465   [.wsbc 2974   [_csb 3069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-sbc 2975  df-csb 3070

This theorem is referenced by:  fsumzcl2  11426  fsumsplitsnun  11440  modfsummodlem1  11477  fprodap0  11642  fprodap0f  11657  fprodmodd  11662  mulcncflem  14361  mulcncf  14362