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Theorem esumeq1d 29424
 Description: Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
Hypotheses
Ref Expression
esumeq1d.0 𝑘𝜑
esumeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
esumeq1d (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)

Proof of Theorem esumeq1d
StepHypRef Expression
1 esumeq1d.0 . 2 𝑘𝜑
2 esumeq1d.1 . 2 (𝜑𝐴 = 𝐵)
3 eqidd 2611 . 2 ((𝜑𝑘𝐴) → 𝐶 = 𝐶)
41, 2, 3esumeq12dvaf 29420 1 (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ 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:  esummono  29443  esumrnmpt2  29457  esumfzf  29458  hasheuni  29474  esum2dlem  29481  measvuni  29604  ddemeas  29626  omssubadd  29689  carsggect  29707
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