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Theorem cbvsumi 11131
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 2281 . 2  |-  F/_ k A
3 nfcv 2281 . 2  |-  F/_ j A
4 cbvsumi.1 . 2  |-  F/_ k B
5 cbvsumi.2 . 2  |-  F/_ j C
61, 2, 3, 4, 5cbvsum 11129 1  |-  sum_ j  e.  A  B  =  sum_ k  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   F/_wnfc 2268   sum_csu 11122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-recs 6202  df-frec 6288  df-seqfrec 10219  df-sumdc 11123
This theorem is referenced by:  sumfct  11143  isumss2  11162  fsumzcl2  11174  fsumsplitf  11177  sumsnf  11178  sumsns  11184  fsumsplitsnun  11188  fsum2dlemstep  11203  fisumcom2  11207  fsumshftm  11214  fsumiun  11246
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