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  8070  mptnn0fsupp  13961  mptnn0fsuppr  13963  fsumzcl2  15684  fsummsnunz  15699  fsumsplitsnun  15700  modfsummodslem1  15737  fprodmodd  15940  sumeven  16329  sumodd  16330  gsummpt1n0  19832  gsummptnn0fz  19853  telgsumfzslem  19855  telgsumfzs  19856  telgsums  19860  mptscmfsupp0  20536  coe1fzgsumdlem  21824  gsummoncoe1  21827  evl1gsumdlem  21874  madugsum  22144  iunmbl2  25073  gsumvsca1  32366  gsumvsca2  32367  rmfsupp2  32382  esum2dlem  33085  esumiun  33087  f1o2d2  41057  iblsplitf  44676  fsummsndifre  46030  fsumsplitsndif  46031  fsummmodsndifre  46032  fsummmodsnunz  46033