Theorem cbvsumv 11382

Description: Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Hypothesis

Ref	Expression
cbvsum.1	|- ( j = k -> B = C )

Assertion

Ref	Expression
cbvsumv	|- sum_ j e. A B = sum_ k e. A C

Distinct variable groups:   A, j, k   B, k   C, j

Allowed substitution hints:   B( j)   C( k)

Proof of Theorem cbvsumv

Step	Hyp	Ref	Expression
1		cbvsum.1	. 2 |- ( j = k -> B = C )
2		nfcv 2329	. 2 |- F/_ k A
3		nfcv 2329	. 2 |- F/_ j A
4		nfcv 2329	. 2 |- F/_ k B
5		nfcv 2329	. 2 |- F/_ j C
6	1, 2, 3, 4, 5	cbvsum 11381	1 |- sum_ j e. A B = sum_ k e. A C

Syntax hints:   -> wi 4   = wceq 1363   sum_csu 11374

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-if 3547  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-recs 6319  df-frec 6405  df-seqfrec 10459  df-sumdc 11375

This theorem is referenced by:  isumge0  11451  telfsumo  11487  fsumparts  11491  binomlem  11504  mertenslemi1  11556  mertenslem2  11557  mertensabs  11558  efaddlem  11695  trilpo  15019  redcwlpo  15031  nconstwlpo  15042  neapmkv  15044