Theorem rspcsbela 4397

Description: Special case related to rspsbc 3839. (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 3839	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → [𝐴 / 𝑥]𝐶 ∈ 𝐷))
2		sbcel1g 4375	. . 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 2109  ∀wral 3044  [wsbc 3750  ⦋csb 3859

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-nul 4293

This theorem is referenced by:  el2mpocsbcl  8041  mptnn0fsupp  13938  mptnn0fsuppr  13940  fsumzcl2  15681  fsummsnunz  15696  fsumsplitsnun  15697  modfsummodslem1  15734  fprodmodd  15939  sumeven  16333  sumodd  16334  gsummpt1n0  19871  gsummptnn0fz  19892  telgsumfzslem  19894  telgsumfzs  19895  telgsums  19899  mptscmfsupp0  20809  coe1fzgsumdlem  22166  gsummoncoe1  22171  evl1gsumdlem  22219  madugsum  22506  iunmbl2  25434  gsumvsca1  33152  gsumvsca2  33153  rmfsupp2  33162  esum2dlem  34055  esumiun  34057  evl1gprodd  42078  idomnnzgmulnz  42094  deg1gprod  42101  f1o2d2  42194  iblsplitf  45941  fsummsndifre  47346  fsumsplitsndif  47347  fsummmodsndifre  47348  fsummmodsnunz  47349