Theorem esumeq2dv 34222

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017.)

Hypothesis

Ref	Expression
esumeq2dv.1	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
esumeq2dv	⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

Distinct variable group:   𝜑,𝑘

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

Proof of Theorem esumeq2dv

Step	Hyp	Ref	Expression
1		nfv 1916	. 2 ⊢ Ⅎ𝑘𝜑
2		esumeq2dv.1	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶)
3	2	ralrimiva 3130	. 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶)
4	1, 3	esumeq2d 34221	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Σ^*cesum 34211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-iota 6458  df-fv 6510  df-ov 7373  df-esum 34212

This theorem is referenced by:  esumeq2sdv  34223  esumle  34242  esummulc1  34265  esummulc2  34266  esumdivc  34267  esumsup  34273  measinb  34405  measres  34406  measdivcst  34408  measdivcstALTV  34409  cntmeas  34410  ddemeas  34420  omsval  34477  totprobd  34610