Theorem rspcsbela 4396

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 4374	. . 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 2145  ∀wral 3076  [wsbc 3739  ⦋csb 3847

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-nul 4280

This theorem is used by:  el2mpocsbcl  8082  mpof1o2d  8123  mptnn0fsupp  14061  mptnn0fsuppr  14063  fsumzcl2  15825  fsummsnunz  15840  fsumsplitsnun  15841  modfsummodslem1  15879  fprodmodd  16084  sumeven  16477  sumodd  16478  gsummpt1n0  20092  gsummptnn0fz  20113  telgsumfzslem  20115  telgsumfzs  20116  telgsums  20120  mptscmfsupp0  21111  coe1fzgsumdlem  22528  gsummoncoe1  22533  evl1gsumdlem  22581  madugsum  22865  iunmbl2  25785  gsummptfzsplitra  33498  gsummptfzsplitla  33499  gsummulsubdishift1s  33510  gsummulsubdishift2s  33511  gsumvsca1  33666  gsumvsca2  33667  rmfsupp2  33677  esum2dlem  34602  esumiun  34604  evl1gprodd  42983  idomnnzgmulnz  42999  deg1gprod  43006  iblsplitf  46798  fsummsndifre  48268  fsumsplitsndif  48269  fsummmodsndifre  48270  fsummmodsnunz  48271