Theorem rspcsbela 4436

Description: Special case related to rspsbc 3874. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)

Assertion

Ref	Expression
rspcsbela	⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 ∈ 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 3874	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → [𝐴 / 𝑥]𝐶 ∈ 𝐷))
2		sbcel1g 4414	. . 3 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ∈ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷))
3	1, 2	sylibd 238	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷))
4	3	imp 408	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2107  ∀wral 3062  [wsbc 3778  ⦋csb 3894

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 3063  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-nul 4324

This theorem is referenced by:  el2mpocsbcl  8071  mptnn0fsupp  13962  mptnn0fsuppr  13964  fsumzcl2  15685  fsummsnunz  15700  fsumsplitsnun  15701  modfsummodslem1  15738  fprodmodd  15941  sumeven  16330  sumodd  16331  gsummpt1n0  19833  gsummptnn0fz  19854  telgsumfzslem  19856  telgsumfzs  19857  telgsums  19861  mptscmfsupp0  20537  coe1fzgsumdlem  21825  gsummoncoe1  21828  evl1gsumdlem  21875  madugsum  22145  iunmbl2  25074  gsumvsca1  32371  gsumvsca2  32372  rmfsupp2  32387  esum2dlem  33090  esumiun  33092  f1o2d2  41055  iblsplitf  44686  fsummsndifre  46040  fsumsplitsndif  46041  fsummmodsndifre  46042  fsummmodsnunz  46043