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Theorem cbvsumv 12495
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  =  k  ->  B  =  C )
Assertion
Ref Expression
cbvsumv  |-  sum_ j  e.  A  B  =  sum_ k  e.  A  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  =  k  ->  B  =  C )
2 nfcv 2574 . 2  |-  F/_ k A
3 nfcv 2574 . 2  |-  F/_ j A
4 nfcv 2574 . 2  |-  F/_ k B
5 nfcv 2574 . 2  |-  F/_ j C
61, 2, 3, 4, 5cbvsum 12494 1  |-  sum_ j  e.  A  B  =  sum_ k  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   sum_csu 12484
This theorem is referenced by:  isumge0  12555  fsumtscopo  12586  fsumparts  12590  binomlem  12613  incexclem  12621  mertenslem1  12666  mertens  12668  efaddlem  12700  bitsinv2  12960  prmreclem6  13294  ovolicc2lem4  19421  uniioombllem6  19485  plymullem1  20138  plyadd  20141  plymul  20142  coeeu  20149  coeid  20162  dvply1  20206  vieta1  20234  aaliou3  20273  abelthlem8  20360  abelthlem9  20361  abelth  20362  logtayl  20556  ftalem2  20861  ftalem6  20865  dchrsum2  21057  sumdchr2  21059  dchrisumlem1  21188  dchrisum  21191  dchrisum0fval  21204  dchrisum0ff  21206  rpvmasum  21225  mulog2sumlem1  21233  2vmadivsumlem  21239  logsqvma  21241  logsqvma2  21242  selberg  21247  chpdifbndlem1  21252  selberg3lem1  21256  selberg4lem1  21259  pntsval  21271  pntsval2  21275  pntpbnd1  21285  pntlemo  21306  hashunif  24163  binomfallfaclem2  25361  axsegconlem9  25869  bpolyval  26100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-recs 6636  df-rdg 6671  df-seq 11329  df-sum 12485
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