Theorem cbvsumi 12072

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

Step	Hyp	Ref	Expression
1		cbvsumi.3	. 2 |- ( j = k -> B = C )
2		nfcv 2386	. 2 |- F/_ k A
3		nfcv 2386	. 2 |- F/_ j A
4		cbvsumi.1	. 2 |- F/_ k B
5		cbvsumi.2	. 2 |- F/_ j C
6	1, 2, 3, 4, 5	cbvsum 12070	1 |- sum_ j e. A B = sum_ k e. A C

Syntax hints:   -> wi 4   = wceq 1398   F/_wnfc 2373   sum_csu 12063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-recs 6549  df-frec 6635  df-seqfrec 10834  df-sumdc 12064

This theorem is referenced by:  sumfct  12084  isumss2  12104  fsumzcl2  12116  fsumsplitf  12119  sumsnf  12120  sumsns  12126  fsumsplitsnun  12130  fsum2dlemstep  12145  fisumcom2  12149  fsumshftm  12156  fsumiun  12188  elplyd  15732  fsumdvdsmul  15985