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Theorem cbvsumv 12169
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 2419 . 2  |-  F/_ k A
3 nfcv 2419 . 2  |-  F/_ j A
4 nfcv 2419 . 2  |-  F/_ k B
5 nfcv 2419 . 2  |-  F/_ j C
61, 2, 3, 4, 5cbvsum 12168 1  |-  sum_ j  e.  A  B  =  sum_ k  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   sum_csu 12158
This theorem is referenced by:  isumge0  12229  fsumtscopo  12260  fsumparts  12264  binomlem  12287  incexclem  12295  mertenslem1  12340  mertens  12342  efaddlem  12374  bitsinv2  12634  prmreclem6  12968  ovolicc2lem4  18879  uniioombllem6  18943  plymullem1  19596  plyadd  19599  plymul  19600  coeeu  19607  coeid  19620  dvply1  19664  vieta1  19692  aaliou3  19731  abelthlem8  19815  abelthlem9  19816  abelth  19817  logtayl  20007  ftalem2  20311  ftalem6  20315  dchrsum2  20507  sumdchr2  20509  dchrisumlem1  20638  dchrisum  20641  dchrisum0fval  20654  dchrisum0ff  20656  rpvmasum  20675  mulog2sumlem1  20683  2vmadivsumlem  20689  logsqvma  20691  logsqvma2  20692  selberg  20697  chpdifbndlem1  20702  selberg3lem1  20706  selberg4lem1  20709  pntsval  20721  pntsval2  20725  pntpbnd1  20735  pntlemo  20756  axsegconlem9  24553  bpolyval  24784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-seq 11047  df-sum 12159
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