Theorem esumeq2sdv 34604

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 486	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶)
3	2	esumeq2dv 34603	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Σ^*cesum 34592

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-10 2178  ax-12 2213  ax-ext 2732

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-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-iota 6483  df-fv 6535  df-ov 7411  df-esum 34593

This theorem is used by:  ismeas  34765  isrnmeas  34766