Theorem rspcsbela 4435

Description: Special case related to rspsbc 3873. (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 3873	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → [𝐴 / 𝑥]𝐶 ∈ 𝐷))
2		sbcel1g 4413	. . 3 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ∈ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷))
3	1, 2	sylibd 238	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷))
4	3	imp 407	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  ∀wral 3061  [wsbc 3777  ⦋csb 3893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-nul 4323

This theorem is referenced by:  el2mpocsbcl  8073  mptnn0fsupp  13964  mptnn0fsuppr  13966  fsumzcl2  15687  fsummsnunz  15702  fsumsplitsnun  15703  modfsummodslem1  15740  fprodmodd  15943  sumeven  16332  sumodd  16333  gsummpt1n0  19835  gsummptnn0fz  19856  telgsumfzslem  19858  telgsumfzs  19859  telgsums  19863  mptscmfsupp0  20542  coe1fzgsumdlem  21832  gsummoncoe1  21835  evl1gsumdlem  21882  madugsum  22152  iunmbl2  25081  gsumvsca1  32412  gsumvsca2  32413  rmfsupp2  32428  esum2dlem  33159  esumiun  33161  f1o2d2  41141  iblsplitf  44765  fsummsndifre  46119  fsumsplitsndif  46120  fsummmodsndifre  46121  fsummmodsnunz  46122