Theorem rspcsbela 4413

Description: Special case related to rspsbc 3854. (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 3854	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → [𝐴 / 𝑥]𝐶 ∈ 𝐷))
2		sbcel1g 4391	. . 3 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ∈ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷))
3	1, 2	sylibd 239	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷))
4	3	imp 406	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∀wral 3051  [wsbc 3765  ⦋csb 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-nul 4309

This theorem is referenced by:  el2mpocsbcl  8084  mptnn0fsupp  14015  mptnn0fsuppr  14017  fsumzcl2  15755  fsummsnunz  15770  fsumsplitsnun  15771  modfsummodslem1  15808  fprodmodd  16013  sumeven  16406  sumodd  16407  gsummpt1n0  19946  gsummptnn0fz  19967  telgsumfzslem  19969  telgsumfzs  19970  telgsums  19974  mptscmfsupp0  20884  coe1fzgsumdlem  22241  gsummoncoe1  22246  evl1gsumdlem  22294  madugsum  22581  iunmbl2  25510  gsumvsca1  33223  gsumvsca2  33224  rmfsupp2  33233  esum2dlem  34123  esumiun  34125  evl1gprodd  42130  idomnnzgmulnz  42146  deg1gprod  42153  f1o2d2  42284  iblsplitf  45999  fsummsndifre  47386  fsumsplitsndif  47387  fsummmodsndifre  47388  fsummmodsnunz  47389