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Theorem esumeq2d 33565
Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016.)
Hypotheses
Ref Expression
esumeq2d.0 𝑘𝜑
esumeq2d.1 (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)
Assertion
Ref Expression
esumeq2d (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)

Proof of Theorem esumeq2d
StepHypRef Expression
1 esumeq2d.0 . 2 𝑘𝜑
2 eqidd 2727 . 2 (𝜑𝐴 = 𝐴)
3 esumeq2d.1 . . 3 (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)
43r19.21bi 3242 . 2 ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
51, 2, 4esumeq12dvaf 33559 1 (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wnf 1777  wral 3055  Σ*cesum 33555
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 6489  df-fv 6545  df-ov 7408  df-esum 33556
This theorem is referenced by:  esumeq2dv  33566  esumpad  33583  esumlef  33590  esumrnmpt2  33596  voliune  33757  omssubadd  33829  carsggect  33847  omsmeas  33852  dstrvprob  34000
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