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Theorem cbvsumi 12186
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 2432 . 2  |-  F/_ k A
3 nfcv 2432 . 2  |-  F/_ j A
4 cbvsumi.1 . 2  |-  F/_ k B
5 cbvsumi.2 . 2  |-  F/_ j C
61, 2, 3, 4, 5cbvsum 12184 1  |-  sum_ j  e.  A  B  =  sum_ k  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   F/_wnfc 2419   sum_csu 12174
This theorem is referenced by:  sumfc  12198  sumss2  12215  sumsn  12229  sumsns  12231  fsum2dlem  12249  fsumcom2  12253  fsumshftm  12259  fsumrlim  12285  fsumo1  12286  o1fsum  12287  fsumiun  12295  ovolfiniun  18876  ovoliun2  18881  volfiniun  18920  itgfsum  19197  elplyd  19600  coeeq2  19640  fsumdvdscom  20441  fsumdvdsmul  20451  fsumvma  20468  hashunif  23400  sumsnd  27800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-seq 11063  df-sum 12175
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