Theorem cbvesumv 33339

Description: Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)

Hypothesis

Ref	Expression
cbvesum.1	⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)

Assertion

Ref	Expression
cbvesumv	⊢ Σ*𝑗 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶

Distinct variable groups:   𝑗,𝑘,𝐴   𝐵,𝑘   𝐶,𝑗

Allowed substitution hints:   𝐵(𝑗)   𝐶(𝑘)

Proof of Theorem cbvesumv

Step	Hyp	Ref	Expression
1		cbvesum.1	. 2 ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)
2		nfcv 2901	. 2 ⊢ Ⅎ𝑘𝐴
3		nfcv 2901	. 2 ⊢ Ⅎ𝑗𝐴
4		nfcv 2901	. 2 ⊢ Ⅎ𝑘𝐵
5		nfcv 2901	. 2 ⊢ Ⅎ𝑗𝐶
6	1, 2, 3, 4, 5	cbvesum 33338	1 ⊢ Σ*𝑗 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶

Syntax hints:   → wi 4   = wceq 1539  Σ^*cesum 33323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-iota 6494  df-fv 6550  df-ov 7414  df-esum 33324

This theorem is referenced by:  esumcvg2  33383  omssubadd  33597  totprob  33724