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Theorem esumeq12dva 32754
Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.) (Revised by Thierry Arnoux, 29-Jun-2017.)
Hypotheses
Ref Expression
esumeq12dva.1 (𝜑𝐴 = 𝐵)
esumeq12dva.2 ((𝜑𝑘𝐴) → 𝐶 = 𝐷)
Assertion
Ref Expression
esumeq12dva (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)
Distinct variable group:   𝜑,𝑘
Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)   𝐶(𝑘)   𝐷(𝑘)

Proof of Theorem esumeq12dva
StepHypRef Expression
1 nfv 1917 . 2 𝑘𝜑
2 esumeq12dva.1 . 2 (𝜑𝐴 = 𝐵)
3 esumeq12dva.2 . 2 ((𝜑𝑘𝐴) → 𝐶 = 𝐷)
41, 2, 3esumeq12dvaf 32753 1 (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Σ*cesum 32749
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3419  df-v 3461  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-opab 5188  df-mpt 5209  df-iota 6468  df-fv 6524  df-ov 7380  df-esum 32750
This theorem is referenced by:  esumeq12d  32755
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