Theorem cbvsumvw2 36428

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 15658	. 2 ⊢ Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐴 𝐷
3		cbvsumvw2.1	. . 3 ⊢ 𝐴 = 𝐵
4	3	sumeq1i 15659	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐷 = Σ𝑘 ∈ 𝐵 𝐷
5	2, 4	eqtri 2759	1 ⊢ Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷

Syntax hints:   → wi 4   = wceq 1542  Σcsu 15648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-xp 5637  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-iota 6454  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-seq 13964  df-sum 15649

This theorem is referenced by: (None)