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Theorem cbvsumv 12445
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 2540 . 2  |-  F/_ k A
3 nfcv 2540 . 2  |-  F/_ j A
4 nfcv 2540 . 2  |-  F/_ k B
5 nfcv 2540 . 2  |-  F/_ j C
61, 2, 3, 4, 5cbvsum 12444 1  |-  sum_ j  e.  A  B  =  sum_ k  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   sum_csu 12434
This theorem is referenced by:  isumge0  12505  fsumtscopo  12536  fsumparts  12540  binomlem  12563  incexclem  12571  mertenslem1  12616  mertens  12618  efaddlem  12650  bitsinv2  12910  prmreclem6  13244  ovolicc2lem4  19369  uniioombllem6  19433  plymullem1  20086  plyadd  20089  plymul  20090  coeeu  20097  coeid  20110  dvply1  20154  vieta1  20182  aaliou3  20221  abelthlem8  20308  abelthlem9  20309  abelth  20310  logtayl  20504  ftalem2  20809  ftalem6  20813  dchrsum2  21005  sumdchr2  21007  dchrisumlem1  21136  dchrisum  21139  dchrisum0fval  21152  dchrisum0ff  21154  rpvmasum  21173  mulog2sumlem1  21181  2vmadivsumlem  21187  logsqvma  21189  logsqvma2  21190  selberg  21195  chpdifbndlem1  21200  selberg3lem1  21204  selberg4lem1  21207  pntsval  21219  pntsval2  21223  pntpbnd1  21233  pntlemo  21254  hashunif  24111  binomfallfaclem2  25307  axsegconlem9  25768  bpolyval  25999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-seq 11279  df-sum 12435
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