Theorem rspcsbela 3188

Description: Special case related to rspsbc 3116. (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 3116	. . 3 |- ( A e. B -> ( A. x e. B C e. D -> [. A / x ]. C e. D ) )
2		sbcel1g 3147	. . 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 2202   A.wral 2511   [.wsbc 3032   [_csb 3128

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-sbc 3033  df-csb 3129

This theorem is referenced by:  fsumzcl2  12046  fsumsplitsnun  12060  modfsummodlem1  12097  fprodap0  12262  fprodap0f  12277  fprodmodd  12282  gsumfzfsumlemm  14683  mulcncflem  15418  mulcncf  15419