Theorem cbvsumvw2 36204

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 = 𝑘 → 𝐶 = 𝐷)

Assertion

Ref	Expression
cbvsumvw2	⊢ Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 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

Step	Hyp	Ref	Expression
1		cbvsumvw2.2	. . 3 ⊢ (𝑗 = 𝑘 → 𝐶 = 𝐷)
2	1	cbvsumv 15738	. 2 ⊢ Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐴 𝐷
3		cbvsumvw2.1	. . 3 ⊢ 𝐴 = 𝐵
4	3	sumeq1i 15739	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐷 = Σ𝑘 ∈ 𝐵 𝐷
5	2, 4	eqtri 2768	1 ⊢ Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷

Syntax hints:   → wi 4   = wceq 1537  Σcsu 15728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-xp 5701  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-pred 6327  df-iota 6520  df-f 6572  df-f1 6573  df-fo 6574  df-f1o 6575  df-fv 6576  df-ov 7446  df-oprab 7447  df-mpo 7448  df-frecs 8316  df-wrecs 8347  df-recs 8421  df-rdg 8460  df-seq 14047  df-sum 15729

This theorem is referenced by: (None)