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Theorem esumeq2dv 32304
Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Hypothesis
Ref Expression
esumeq2dv.1 ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
esumeq2dv (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Distinct variable group:   𝜑,𝑘
Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)   𝐶(𝑘)

Proof of Theorem esumeq2dv
StepHypRef Expression
1 nfv 1916 . 2 𝑘𝜑
2 esumeq2dv.1 . . 3 ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
32ralrimiva 3139 . 2 (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)
41, 3esumeq2d 32303 1 (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  Σ*cesum 32293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-iota 6431  df-fv 6487  df-ov 7340  df-esum 32294
This theorem is referenced by:  esumeq2sdv  32305  esumle  32324  esummulc1  32347  esummulc2  32348  esumdivc  32349  esumsup  32355  measinb  32487  measres  32488  measdivcst  32490  measdivcstALTV  32491  cntmeas  32492  ddemeas  32502  omsval  32560  totprobd  32693
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