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Theorem rspcsbela 3059
 Description: Special case related to rspsbc 2991. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
Assertion
Ref Expression
rspcsbela ((𝐴𝐵 ∧ ∀𝑥𝐵 𝐶𝐷) → 𝐴 / 𝑥𝐶𝐷)
Distinct variable groups:   𝑥,𝐵   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem rspcsbela
StepHypRef Expression
1 rspsbc 2991 . . 3 (𝐴𝐵 → (∀𝑥𝐵 𝐶𝐷[𝐴 / 𝑥]𝐶𝐷))
2 sbcel1g 3021 . . 3 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐷))
31, 2sylibd 148 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝐶𝐷𝐴 / 𝑥𝐶𝐷))
43imp 123 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝐶𝐷) → 𝐴 / 𝑥𝐶𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1480  ∀wral 2416  [wsbc 2909  ⦋csb 3003 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-sbc 2910  df-csb 3004 This theorem is referenced by:  fsumzcl2  11181  fsumsplitsnun  11195  modfsummodlem1  11232  mulcncflem  12769  mulcncf  12770
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