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Theorem rspcsbela 3984
Description: Special case related to rspsbc 3504. (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 3504 . . 3 (𝐴𝐵 → (∀𝑥𝐵 𝐶𝐷[𝐴 / 𝑥]𝐶𝐷))
2 sbcel1g 3965 . . 3 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐷))
31, 2sylibd 229 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝐶𝐷𝐴 / 𝑥𝐶𝐷))
43imp 445 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝐶𝐷) → 𝐴 / 𝑥𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2908  [wsbc 3422  csb 3519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-nul 3898
This theorem is referenced by:  el2mpt2csbcl  7210  mptnn0fsupp  12753  mptnn0fsuppr  12755  fsumzcl2  14418  fsummsnunz  14432  fsumsplitsnun  14433  modfsummodslem1  14470  fprodmodd  14672  sumeven  15053  sumodd  15054  gsummpt1n0  18304  gsummptnn0fz  18322  telgsumfzslem  18325  telgsumfzs  18326  telgsums  18330  mptscmfsupp0  18868  coe1fzgsumdlem  19611  gsummoncoe1  19614  evl1gsumdlem  19660  madugsum  20389  iunmbl2  23265  gsumvsca1  29609  gsumvsca2  29610  esum2dlem  29977  esumiun  29979  iblsplitf  39523  fsummsndifre  40670  fsumsplitsndif  40671  fsummmodsndifre  40672  fsummmodsnunz  40673
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