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Theorem cbvesumv 31990
Description: Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)
Hypothesis
Ref Expression
cbvesum.1 (𝑗 = 𝑘𝐵 = 𝐶)
Assertion
Ref Expression
cbvesumv Σ*𝑗𝐴𝐵 = Σ*𝑘𝐴𝐶
Distinct variable groups:   𝑗,𝑘,𝐴   𝐵,𝑘   𝐶,𝑗
Allowed substitution hints:   𝐵(𝑗)   𝐶(𝑘)

Proof of Theorem cbvesumv
StepHypRef Expression
1 cbvesum.1 . 2 (𝑗 = 𝑘𝐵 = 𝐶)
2 nfcv 2908 . 2 𝑘𝐴
3 nfcv 2908 . 2 𝑗𝐴
4 nfcv 2908 . 2 𝑘𝐵
5 nfcv 2908 . 2 𝑗𝐶
61, 2, 3, 4, 5cbvesum 31989 1 Σ*𝑗𝐴𝐵 = Σ*𝑘𝐴𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  Σ*cesum 31974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-iota 6388  df-fv 6438  df-ov 7271  df-esum 31975
This theorem is referenced by:  esumcvg2  32034  omssubadd  32246  totprob  32373
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