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Theorem esumeq2d 31353
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 2825 . 2 (𝜑𝐴 = 𝐴)
3 esumeq2d.1 . . 3 (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)
43r19.21bi 3203 . 2 ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
51, 2, 4esumeq12dvaf 31347 1 (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wnf 1785  wral 3133  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:  esumeq2dv  31354  esumpad  31371  esumlef  31378  esumrnmpt2  31384  voliune  31545  omssubadd  31615  carsggect  31633  omsmeas  31638  dstrvprob  31786
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