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Theorem rspcsbela 4346
Description: Special case related to rspsbc 3811. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
Assertion
Ref Expression
rspcsbela ((𝐴𝐵 ∧ ∀𝑥𝐵 𝐶𝐷) → 𝐴 / 𝑥𝐶𝐷)
Distinct variable groups:   𝑥,𝐵   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem rspcsbela
StepHypRef Expression
1 rspsbc 3811 . . 3 (𝐴𝐵 → (∀𝑥𝐵 𝐶𝐷[𝐴 / 𝑥]𝐶𝐷))
2 sbcel1g 4324 . . 3 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐷))
31, 2sylibd 242 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝐶𝐷𝐴 / 𝑥𝐶𝐷))
43imp 410 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝐶𝐷) → 𝐴 / 𝑥𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2112  wral 3109  [wsbc 3723  csb 3831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-nul 4247
This theorem is referenced by:  el2mpocsbcl  7767  mptnn0fsupp  13364  mptnn0fsuppr  13366  fsumzcl2  15091  fsummsnunz  15105  fsumsplitsnun  15106  modfsummodslem1  15143  fprodmodd  15347  sumeven  15732  sumodd  15733  gsummpt1n0  19082  gsummptnn0fz  19103  telgsumfzslem  19105  telgsumfzs  19106  telgsums  19110  mptscmfsupp0  19696  coe1fzgsumdlem  20934  gsummoncoe1  20937  evl1gsumdlem  20984  madugsum  21252  iunmbl2  24165  gsumvsca1  30908  gsumvsca2  30909  rmfsupp2  30921  esum2dlem  31465  esumiun  31467  iblsplitf  42609  fsummsndifre  43886  fsumsplitsndif  43887  fsummmodsndifre  43888  fsummmodsnunz  43889
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