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Theorem esumeq12dva 31365
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 1915 . 2 𝑘𝜑
2 esumeq12dva.1 . 2 (𝜑𝐴 = 𝐵)
3 esumeq12dva.2 . 2 ((𝜑𝑘𝐴) → 𝐶 = 𝐷)
41, 2, 3esumeq12dvaf 31364 1 (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  Σ*cesum 31360
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 2116  ax-9 2124  ax-10 2145  ax-12 2178  ax-ext 2794
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 2801  df-cleq 2815  df-clel 2894  df-ral 3135  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-iota 6293  df-fv 6342  df-ov 7143  df-esum 31361
This theorem is referenced by:  esumeq12d  31366
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