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Theorem esumeq12d 31191
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 481 . 2 ((𝜑𝑘𝐴) → 𝐶 = 𝐷)
41, 3esumeq12dva 31190 1 (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Σ*cesum 31185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-iota 6307  df-fv 6356  df-ov 7148  df-esum 31186
This theorem is referenced by:  esumeq1  31192  esum2dlem  31250
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