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Theorem cbvsumv 15055
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 2979 . 2 𝑘𝐴
3 nfcv 2979 . 2 𝑗𝐴
4 nfcv 2979 . 2 𝑘𝐵
5 nfcv 2979 . 2 𝑗𝐶
61, 2, 3, 4, 5cbvsum 15054 1 Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  Σcsu 15044
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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-xp 5563  df-cnv 5565  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-iota 6316  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-seq 13373  df-sum 15045
This theorem is referenced by:  isumge0  15123  telfsumo  15159  fsumparts  15163  binomlem  15186  incexclem  15193  pwdif  15225  mertenslem1  15242  mertens  15244  binomfallfaclem2  15396  bpolyval  15405  efaddlem  15448  pwp1fsum  15744  bitsinv2  15794  prmreclem6  16259  ovolicc2lem4  24123  uniioombllem6  24191  plymullem1  24806  plyadd  24809  plymul  24810  coeeu  24817  coeid  24830  dvply1  24875  vieta1  24903  aaliou3  24942  abelthlem8  25029  abelthlem9  25030  abelth  25031  logtayl  25245  ftalem2  25653  ftalem6  25657  dchrsum2  25846  sumdchr2  25848  dchrisumlem1  26067  dchrisum  26070  dchrisum0fval  26083  dchrisum0ff  26085  rpvmasum  26104  mulog2sumlem1  26112  2vmadivsumlem  26118  logsqvma  26120  logsqvma2  26121  selberg  26126  chpdifbndlem1  26131  selberg3lem1  26135  selberg4lem1  26138  pntsval  26150  pntsval2  26154  pntpbnd1  26164  pntlemo  26185  axsegconlem9  26713  hashunif  30530  eulerpartlems  31620  eulerpartlemgs2  31640  breprexplema  31903  breprexplemc  31905  breprexp  31906  hgt750lema  31930  fwddifnp1  33628  binomcxplemnotnn0  40695  mccl  41886  sumnnodd  41918  dvnprodlem1  42238  dvnprodlem3  42240  dvnprod  42241  fourierdlem73  42471  fourierdlem112  42510  fourierdlem113  42511  etransclem11  42537  etransclem32  42558  etransclem35  42561  etransc  42575  fsumlesge0  42666  meaiuninclem  42769  omeiunltfirp  42808  hoidmvlelem3  42886  altgsumbcALT  44408  nn0sumshdiglemA  44686  nn0sumshdiglemB  44687
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