Theorem rspcsbela 4401

Description: Special case related to rspsbc 3842. (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 3842	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → [𝐴 / 𝑥]𝐶 ∈ 𝐷))
2		sbcel1g 4379	. . 3 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ∈ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷))
3	1, 2	sylibd 239	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷))
4	3	imp 406	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3044  [wsbc 3753  ⦋csb 3862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-nul 4297

This theorem is referenced by:  el2mpocsbcl  8064  mptnn0fsupp  13962  mptnn0fsuppr  13964  fsumzcl2  15705  fsummsnunz  15720  fsumsplitsnun  15721  modfsummodslem1  15758  fprodmodd  15963  sumeven  16357  sumodd  16358  gsummpt1n0  19895  gsummptnn0fz  19916  telgsumfzslem  19918  telgsumfzs  19919  telgsums  19923  mptscmfsupp0  20833  coe1fzgsumdlem  22190  gsummoncoe1  22195  evl1gsumdlem  22243  madugsum  22530  iunmbl2  25458  gsumvsca1  33179  gsumvsca2  33180  rmfsupp2  33189  esum2dlem  34082  esumiun  34084  evl1gprodd  42105  idomnnzgmulnz  42121  deg1gprod  42128  f1o2d2  42221  iblsplitf  45968  fsummsndifre  47373  fsumsplitsndif  47374  fsummmodsndifre  47375  fsummmodsnunz  47376