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Theorem esumeq1d 33562
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 2727 . 2 ((𝜑𝑘𝐴) → 𝐶 = 𝐶)
41, 2, 3esumeq12dvaf 33558 1 (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wnf 1777  wcel 2098  Σ*cesum 33554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-iota 6488  df-fv 6544  df-ov 7407  df-esum 33555
This theorem is referenced by:  esummono  33581  esumrnmpt2  33595  esumfzf  33596  hasheuni  33612  esum2dlem  33619  measvuni  33741  ddemeas  33763  omssubadd  33828  carsggect  33846
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