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Theorem rspcsbela 2987
Description: Special case related to rspsbc 2921. (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 2921 . . 3 (𝐴𝐵 → (∀𝑥𝐵 𝐶𝐷[𝐴 / 𝑥]𝐶𝐷))
2 sbcel1g 2950 . . 3 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐷))
31, 2sylibd 147 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝐶𝐷𝐴 / 𝑥𝐶𝐷))
43imp 122 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝐶𝐷) → 𝐴 / 𝑥𝐶𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1438  wral 2359  [wsbc 2840  csb 2933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-sbc 2841  df-csb 2934
This theorem is referenced by:  fsumzcl2  10795  fsumsplitsnun  10809  modfsummodlem1  10846
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