Theorem rspcsbela 3858

Description: Special case related to rspsbc 3413. (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 3413	. . 3 |- ( A e. B -> ( A. x e. B C e. D -> [. A / x ]. C e. D ) )
2		sbcel1g 3837	. . 3 |- ( A e. B -> ( [. A / x ]. C e. D <-> [_ A / x ]_ C e. D ) )
3	1, 2	sylibd 214	. 2 |- ( A e. B -> ( A. x e. B C e. D -> [_ A / x ]_ C e. D ) )
4	3	imp 429	1 |- ( ( A e. B /\ A. x e. B C e. D ) -> [_ A / x ]_ C e. D )

Syntax hints:   -> wi 4   /\ wa 369   e. wcel 1819   A.wral 2807   [.wsbc 3327   [_csb 3430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794

This theorem is referenced by:  mptnn0fsupp  12106  mptnn0fsuppr  12108  fsumzcl2  13572  fsummsnunz  13581  fsumsplitsnun  13582  modfsummodslem1  13618  gsummpt1n0  17119  gsummptnn0fz  17141  telgsumfzslem  17144  telgsumfzs  17145  telgsums  17149  mptscmfsupp0  17703  coe1fzgsumdlem  18470  gsummoncoe1  18473  evl1gsumdlem  18519  madugsum  19272  iunmbl2  22093  gsumvsca1  27934  gsumvsca2  27935  esum2dlem  28264  esumiun  28266  iblsplitf  31972  fsummsndifre  32607  fsumsplitsndif  32608  fsummmodsndifre  32609  fsummmodsnunz  32610