Theorem cbvsumvw2 36567

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 15714	. 2 ⊢ Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐴 𝐷
3		cbvsumvw2.1	. . 3 ⊢ 𝐴 = 𝐵
4	3	sumeq1i 15715	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐷 = Σ𝑘 ∈ 𝐵 𝐷
5	2, 4	eqtri 2784	1 ⊢ Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷

Syntax hints:   → wi 4   = wceq 1559  Σcsu 15704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-xp 5649  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-iota 6472  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-seq 14009  df-sum 15705

This theorem is referenced by: (None)