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Theorem cbvsumi 12491
Description: Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.)
Hypotheses
Ref Expression
cbvsumi.1  |-  F/_ k B
cbvsumi.2  |-  F/_ j C
cbvsumi.3  |-  ( j  =  k  ->  B  =  C )
Assertion
Ref Expression
cbvsumi  |-  sum_ j  e.  A  B  =  sum_ k  e.  A  C
Distinct variable group:    j, k, A
Allowed substitution hints:    B( j, k)    C( j, k)

Proof of Theorem cbvsumi
StepHypRef Expression
1 cbvsumi.3 . 2  |-  ( j  =  k  ->  B  =  C )
2 nfcv 2572 . 2  |-  F/_ k A
3 nfcv 2572 . 2  |-  F/_ j A
4 cbvsumi.1 . 2  |-  F/_ k B
5 cbvsumi.2 . 2  |-  F/_ j C
61, 2, 3, 4, 5cbvsum 12489 1  |-  sum_ j  e.  A  B  =  sum_ k  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   F/_wnfc 2559   sum_csu 12479
This theorem is referenced by:  sumfc  12503  sumss2  12520  sumsn  12534  sumsns  12536  fsum2dlem  12554  fsumcom2  12558  fsumshftm  12564  fsumrlim  12590  fsumo1  12591  o1fsum  12592  fsumiun  12600  ovolfiniun  19397  ovoliun2  19402  volfiniun  19441  itgfsum  19718  elplyd  20121  coeeq2  20161  fsumdvdscom  20970  fsumdvdsmul  20980  fsumvma  20997  sumsnd  27673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-seq 11324  df-sum 12480
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