Theorem rspcsbela 4399

Description: Special case related to rspsbc 3830. (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 3830	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → [𝐴 / 𝑥]𝐶 ∈ 𝐷))
2		sbcel1g 4377	. . 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 2147  ∀wral 3080  [wsbc 3742  ⦋csb 3851

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-v 3456  df-sbc 3743  df-csb 3852  df-dif 4050  df-nul 4282

This theorem is used by:  el2mpocsbcl  8131  mpof1o2d  8174  mptnn0fsupp  14180  mptnn0fsuppr  14182  fsumzcl2  15949  fsummsnunz  15964  fsumsplitsnun  15965  modfsummodslem1  16003  fprodmodd  16208  sumeven  16601  sumodd  16602  gsummpt1n0  20230  gsummptnn0fz  20251  telgsumfzslem  20253  telgsumfzs  20254  telgsums  20258  mptscmfsupp0  21253  coe1fzgsumdlem  22676  gsummoncoe1  22681  evl1gsumdlem  22729  madugsum  23013  iunmbl2  25933  gsummptfzsplitra  33718  gsummptfzsplitla  33719  gsummulsubdishift1s  33730  gsummulsubdishift2s  33731  gsumvsca1  33886  gsumvsca2  33887  rmfsupp2  33897  esum2dlem  34835  esumiun  34837  evl1gprodd  43433  idomnnzgmulnz  43449  deg1gprod  43456  iblsplitf  47253  fsummsndifre  48723  fsumsplitsndif  48724  fsummmodsndifre  48725  fsummmodsnunz  48726