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Theorem esumeq2dv 29427
 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 1830 . 2 𝑘𝜑
2 esumeq2dv.1 . . 3 ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
32ralrimiva 2949 . 2 (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)
41, 3esumeq2d 29426 1 (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Σ*cesum 29416 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-iota 5768  df-fv 5812  df-ov 6552  df-esum 29417 This theorem is referenced by:  esumeq2sdv  29428  esumle  29447  esummulc1  29470  esummulc2  29471  esumdivc  29472  esumsup  29478  measinb  29611  measres  29612  measdivcstOLD  29614  measdivcst  29615  cntmeas  29616  ddemeas  29626  omsval  29682  totprobd  29815
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