Theorem rspcsbela 4383

Description: Special case related to rspsbc 3825. (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 3825	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → [𝐴 / 𝑥]𝐶 ∈ 𝐷))
2		sbcel1g 4361	. . 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 2111  ∀wral 3047  [wsbc 3736  ⦋csb 3845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-nul 4279

This theorem is referenced by:  el2mpocsbcl  8010  mptnn0fsupp  13899  mptnn0fsuppr  13901  fsumzcl2  15641  fsummsnunz  15656  fsumsplitsnun  15657  modfsummodslem1  15694  fprodmodd  15899  sumeven  16293  sumodd  16294  gsummpt1n0  19872  gsummptnn0fz  19893  telgsumfzslem  19895  telgsumfzs  19896  telgsums  19900  mptscmfsupp0  20855  coe1fzgsumdlem  22213  gsummoncoe1  22218  evl1gsumdlem  22266  madugsum  22553  iunmbl2  25480  gsumvsca1  33187  gsumvsca2  33188  rmfsupp2  33197  esum2dlem  34097  esumiun  34099  evl1gprodd  42150  idomnnzgmulnz  42166  deg1gprod  42173  f1o2d2  42266  iblsplitf  46008  fsummsndifre  47403  fsumsplitsndif  47404  fsummmodsndifre  47405  fsummmodsnunz  47406