Theorem cbvsumvw2 37015

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 15856	. 2 ⊢ Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐴 𝐷
3		cbvsumvw2.1	. . 3 ⊢ 𝐴 = 𝐵
4	3	sumeq1i 15857	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐷 = Σ𝑘 ∈ 𝐵 𝐷
5	2, 4	eqtri 2784	1 ⊢ Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Σcsu 15846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-xp 5657  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-iota 6493  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-seq 14138  df-sum 15847

This theorem is used by: (None)