Theorem esumeq2sdv 34438

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 25-Dec-2016.)

Hypothesis

Ref	Expression
esumeq2sdv.1	⊢ (𝜑 → 𝐵 = 𝐶)

Assertion

Ref	Expression
esumeq2sdv	⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

Distinct variable group:   𝜑,𝑘

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

Proof of Theorem esumeq2sdv

Step	Hyp	Ref	Expression
1		esumeq2sdv.1	. . 3 ⊢ (𝜑 → 𝐵 = 𝐶)
2	1	adantr 485	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶)
3	2	esumeq2dv 34437	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  Σ^*cesum 34426

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-iota 6492  df-fv 6544  df-ov 7415  df-esum 34427

This theorem is used by:  ismeas  34598  isrnmeas  34599