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Theorem cbvsumi 12170
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 2419 . 2  |-  F/_ k A
3 nfcv 2419 . 2  |-  F/_ j A
4 cbvsumi.1 . 2  |-  F/_ k B
5 cbvsumi.2 . 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   F/_wnfc 2406   sum_csu 12158
This theorem is referenced by:  sumfc  12182  sumss2  12199  sumsn  12213  sumsns  12215  fsum2dlem  12233  fsumcom2  12237  fsumshftm  12243  fsumrlim  12269  fsumo1  12270  o1fsum  12271  fsumiun  12279  ovolfiniun  18860  ovoliun2  18865  volfiniun  18904  itgfsum  19181  elplyd  19584  coeeq2  19624  fsumdvdscom  20425  fsumdvdsmul  20435  fsumvma  20452  hashunif  23385  sumsnd  27697
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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