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Theorem esumeq12d 33685
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 479 . 2 ((𝜑𝑘𝐴) → 𝐶 = 𝐷)
41, 3esumeq12dva 33684 1 (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Σ*cesum 33679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-iota 6505  df-fv 6561  df-ov 7429  df-esum 33680
This theorem is referenced by:  esumeq1  33686  esum2dlem  33744
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