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Theorem esumeq2dv 34019
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 1912 . 2 𝑘𝜑
2 esumeq2dv.1 . . 3 ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
32ralrimiva 3144 . 2 (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)
41, 3esumeq2d 34018 1 (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Σ*cesum 34008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-iota 6516  df-fv 6571  df-ov 7434  df-esum 34009
This theorem is referenced by:  esumeq2sdv  34020  esumle  34039  esummulc1  34062  esummulc2  34063  esumdivc  34064  esumsup  34070  measinb  34202  measres  34203  measdivcst  34205  measdivcstALTV  34206  cntmeas  34207  ddemeas  34217  omsval  34275  totprobd  34408
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