Theorem rspcsbela 4438

Description: Special case related to rspsbc 3879. (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 3879	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → [𝐴 / 𝑥]𝐶 ∈ 𝐷))
2		sbcel1g 4416	. . 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 2108  ∀wral 3061  [wsbc 3788  ⦋csb 3899

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-nul 4334

This theorem is referenced by:  el2mpocsbcl  8110  mptnn0fsupp  14038  mptnn0fsuppr  14040  fsumzcl2  15775  fsummsnunz  15790  fsumsplitsnun  15791  modfsummodslem1  15828  fprodmodd  16033  sumeven  16424  sumodd  16425  gsummpt1n0  19983  gsummptnn0fz  20004  telgsumfzslem  20006  telgsumfzs  20007  telgsums  20011  mptscmfsupp0  20925  coe1fzgsumdlem  22307  gsummoncoe1  22312  evl1gsumdlem  22360  madugsum  22649  iunmbl2  25592  gsumvsca1  33232  gsumvsca2  33233  rmfsupp2  33242  esum2dlem  34093  esumiun  34095  evl1gprodd  42118  idomnnzgmulnz  42134  deg1gprod  42141  f1o2d2  42274  iblsplitf  45985  fsummsndifre  47359  fsumsplitsndif  47360  fsummmodsndifre  47361  fsummmodsnunz  47362