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Theorem cbvsumv 14617
Description: Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
Hypothesis
Ref Expression
cbvsum.1 (𝑗 = 𝑘𝐵 = 𝐶)
Assertion
Ref Expression
cbvsumv Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶
Distinct variable groups:   𝑗,𝑘,𝐴   𝐵,𝑘   𝐶,𝑗
Allowed substitution hints:   𝐵(𝑗)   𝐶(𝑘)

Proof of Theorem cbvsumv
StepHypRef Expression
1 cbvsum.1 . 2 (𝑗 = 𝑘𝐵 = 𝐶)
2 nfcv 2894 . 2 𝑘𝐴
3 nfcv 2894 . 2 𝑗𝐴
4 nfcv 2894 . 2 𝑘𝐵
5 nfcv 2894 . 2 𝑗𝐶
61, 2, 3, 4, 5cbvsum 14616 1 Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1624  Σcsu 14607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-xp 5264  df-cnv 5266  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-iota 6004  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-seq 12988  df-sum 14608
This theorem is referenced by:  isumge0  14688  telfsumo  14725  fsumparts  14729  binomlem  14752  incexclem  14759  mertenslem1  14807  mertens  14809  binomfallfaclem2  14962  bpolyval  14971  efaddlem  15014  pwp1fsum  15308  bitsinv2  15359  prmreclem6  15819  ovolicc2lem4  23480  uniioombllem6  23548  plymullem1  24161  plyadd  24164  plymul  24165  coeeu  24172  coeid  24185  dvply1  24230  vieta1  24258  aaliou3  24297  abelthlem8  24384  abelthlem9  24385  abelth  24386  logtayl  24597  ftalem2  24991  ftalem6  24995  dchrsum2  25184  sumdchr2  25186  dchrisumlem1  25369  dchrisum  25372  dchrisum0fval  25385  dchrisum0ff  25387  rpvmasum  25406  mulog2sumlem1  25414  2vmadivsumlem  25420  logsqvma  25422  logsqvma2  25423  selberg  25428  chpdifbndlem1  25433  selberg3lem1  25437  selberg4lem1  25440  pntsval  25452  pntsval2  25456  pntpbnd1  25466  pntlemo  25487  axsegconlem9  25996  hashunif  29863  eulerpartlems  30723  eulerpartlemgs2  30743  breprexplema  31009  breprexplemc  31011  breprexp  31012  hgt750lema  31036  fwddifnp1  32570  binomcxplemnotnn0  39049  mccl  40325  sumnnodd  40357  dvnprodlem1  40656  dvnprodlem3  40658  dvnprod  40659  fourierdlem73  40891  fourierdlem112  40930  fourierdlem113  40931  etransclem11  40957  etransclem32  40978  etransclem35  40981  etransc  40995  fsumlesge0  41089  meaiuninclem  41192  omeiunltfirp  41231  hoidmvlelem3  41309  pwdif  42003  altgsumbcALT  42633  nn0sumshdiglemA  42915  nn0sumshdiglemB  42916
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