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Theorem rspcsbela 4396
Description: Special case related to rspsbc 3836. (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 3836 . . 3 (𝐴𝐵 → (∀𝑥𝐵 𝐶𝐷[𝐴 / 𝑥]𝐶𝐷))
2 sbcel1g 4374 . . 3 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐷))
31, 2sylibd 238 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝐶𝐷𝐴 / 𝑥𝐶𝐷))
43imp 408 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝐶𝐷) → 𝐴 / 𝑥𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wral 3061  [wsbc 3740  csb 3856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-nul 4284
This theorem is referenced by:  el2mpocsbcl  8018  mptnn0fsupp  13908  mptnn0fsuppr  13910  fsumzcl2  15629  fsummsnunz  15644  fsumsplitsnun  15645  modfsummodslem1  15682  fprodmodd  15885  sumeven  16274  sumodd  16275  gsummpt1n0  19747  gsummptnn0fz  19768  telgsumfzslem  19770  telgsumfzs  19771  telgsums  19775  mptscmfsupp0  20402  coe1fzgsumdlem  21688  gsummoncoe1  21691  evl1gsumdlem  21738  madugsum  22008  iunmbl2  24937  gsumvsca1  32110  gsumvsca2  32111  rmfsupp2  32122  esum2dlem  32748  esumiun  32750  iblsplitf  44297  fsummsndifre  45650  fsumsplitsndif  45651  fsummmodsndifre  45652  fsummmodsnunz  45653
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