Theorem rspcsbela 4395

Description: Special case related to rspsbc 3826. (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 3826	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → [𝐴 / 𝑥]𝐶 ∈ 𝐷))
2		sbcel1g 4373	. . 3 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ∈ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷))
3	1, 2	sylibd 242	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷))
4	3	imp 412	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144  ∀wral 3076  [wsbc 3738  ⦋csb 3847

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-v 3452  df-sbc 3739  df-csb 3848  df-dif 4046  df-nul 4278

This theorem is used by:  el2mpocsbcl  8098  mpof1o2d  8141  mptnn0fsupp  14145  mptnn0fsuppr  14147  fsumzcl2  15912  fsummsnunz  15927  fsumsplitsnun  15928  modfsummodslem1  15966  fprodmodd  16171  sumeven  16564  sumodd  16565  gsummpt1n0  20186  gsummptnn0fz  20207  telgsumfzslem  20209  telgsumfzs  20210  telgsums  20214  mptscmfsupp0  21209  coe1fzgsumdlem  22628  gsummoncoe1  22633  evl1gsumdlem  22681  madugsum  22965  iunmbl2  25885  gsummptfzsplitra  33667  gsummptfzsplitla  33668  gsummulsubdishift1s  33679  gsummulsubdishift2s  33680  gsumvsca1  33835  gsumvsca2  33836  rmfsupp2  33846  esum2dlem  34772  esumiun  34774  evl1gprodd  43274  idomnnzgmulnz  43290  deg1gprod  43297  iblsplitf  47087  fsummsndifre  48557  fsumsplitsndif  48558  fsummmodsndifre  48559  fsummmodsnunz  48560