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Theorem cbvsumvw2 36487
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 15653 . 2 Σ𝑗𝐴 𝐶 = Σ𝑘𝐴 𝐷
3 cbvsumvw2.1 . . 3 𝐴 = 𝐵
43sumeq1i 15654 . 2 Σ𝑘𝐴 𝐷 = Σ𝑘𝐵 𝐷
52, 4eqtri 2764 1 Σ𝑗𝐴 𝐶 = Σ𝑘𝐵 𝐷
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1548  Σcsu 15643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713
This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-xp 5626  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-iota 6444  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-seq 13959  df-sum 15644
This theorem is referenced by: (None)
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