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Theorem esumeq1d 31351
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 2825 . 2 ((𝜑𝑘𝐴) → 𝐶 = 𝐶)
41, 2, 3esumeq12dvaf 31347 1 (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wnf 1785  wcel 2115  Σ*cesum 31343
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2179  ax-ext 2796
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ral 3138  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-iota 6302  df-fv 6351  df-ov 7152  df-esum 31344
This theorem is referenced by:  esummono  31370  esumrnmpt2  31384  esumfzf  31385  hasheuni  31401  esum2dlem  31408  measvuni  31530  ddemeas  31552  omssubadd  31615  carsggect  31633
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