Theorem rspcsbela 3118

Description: Special case related to rspsbc 3047. (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 3047	. . 3 |- ( A e. B -> ( A. x e. B C e. D -> [. A / x ]. C e. D ) )
2		sbcel1g 3078	. . 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 2148   A.wral 2455   [.wsbc 2964   [_csb 3059

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-sbc 2965  df-csb 3060

This theorem is referenced by:  fsumzcl2  11415  fsumsplitsnun  11429  modfsummodlem1  11466  fprodap0  11631  fprodap0f  11646  fprodmodd  11651  mulcncflem  14175  mulcncf  14176