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Theorem cbvsumvw2 36225
Description: Change bound variable and the set of integers in a sum, using implicit substitution. (Contributed by GG, 1-Sep-2025.)
Hypotheses
Ref Expression
cbvsumvw2.1 𝐴 = 𝐵
cbvsumvw2.2 (𝑗 = 𝑘𝐶 = 𝐷)
Assertion
Ref Expression
cbvsumvw2 Σ𝑗𝐴 𝐶 = Σ𝑘𝐵 𝐷
Distinct variable groups:   𝑗,𝑘   𝐷,𝑗   𝐶,𝑘
Allowed substitution hints:   𝐴(𝑗,𝑘)   𝐵(𝑗,𝑘)   𝐶(𝑗)   𝐷(𝑘)

Proof of Theorem cbvsumvw2
StepHypRef Expression
1 cbvsumvw2.2 . . 3 (𝑗 = 𝑘𝐶 = 𝐷)
21cbvsumv 15728 . 2 Σ𝑗𝐴 𝐶 = Σ𝑘𝐴 𝐷
3 cbvsumvw2.1 . . 3 𝐴 = 𝐵
43sumeq1i 15729 . 2 Σ𝑘𝐴 𝐷 = Σ𝑘𝐵 𝐷
52, 4eqtri 2764 1 Σ𝑗𝐴 𝐶 = Σ𝑘𝐵 𝐷
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  Σcsu 15718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-xp 5689  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-iota 6512  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-seq 14039  df-sum 15719
This theorem is referenced by: (None)
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