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Theorem esumeq2dv 31411
Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Hypothesis
Ref Expression
esumeq2dv.1 ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
esumeq2dv (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Distinct variable group:   𝜑,𝑘
Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)   𝐶(𝑘)

Proof of Theorem esumeq2dv
StepHypRef Expression
1 nfv 1915 . 2 𝑘𝜑
2 esumeq2dv.1 . . 3 ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
32ralrimiva 3152 . 2 (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)
41, 3esumeq2d 31410 1 (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  Σ*cesum 31400
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-iota 6287  df-fv 6336  df-ov 7142  df-esum 31401
This theorem is referenced by:  esumeq2sdv  31412  esumle  31431  esummulc1  31454  esummulc2  31455  esumdivc  31456  esumsup  31462  measinb  31594  measres  31595  measdivcst  31597  measdivcstALTV  31598  cntmeas  31599  ddemeas  31609  omsval  31665  totprobd  31798
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