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Theorem esumeq12d 31468
 Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
Hypotheses
Ref Expression
esumeq12d.1 (𝜑𝐴 = 𝐵)
esumeq12d.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
esumeq12d (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)
Distinct variable group:   𝜑,𝑘
Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)   𝐶(𝑘)   𝐷(𝑘)

Proof of Theorem esumeq12d
StepHypRef Expression
1 esumeq12d.1 . 2 (𝜑𝐴 = 𝐵)
2 esumeq12d.2 . . 3 (𝜑𝐶 = 𝐷)
32adantr 484 . 2 ((𝜑𝑘𝐴) → 𝐶 = 𝐷)
41, 3esumeq12dva 31467 1 (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Σ*cesum 31462 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770 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 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3444  df-un 3888  df-in 3890  df-ss 3900  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-iota 6291  df-fv 6340  df-ov 7148  df-esum 31463 This theorem is referenced by:  esumeq1  31469  esum2dlem  31527
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