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Theorem esumeq2sdv 33789
Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Hypothesis
Ref Expression
esumeq2sdv.1 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
esumeq2sdv (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Distinct variable group:   𝜑,𝑘
Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)   𝐶(𝑘)

Proof of Theorem esumeq2sdv
StepHypRef Expression
1 esumeq2sdv.1 . . 3 (𝜑𝐵 = 𝐶)
21adantr 479 . 2 ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
32esumeq2dv 33788 1 (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Σ*cesum 33777
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 2696
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 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-iota 6501  df-fv 6557  df-ov 7422  df-esum 33778
This theorem is referenced by:  ismeas  33949  isrnmeas  33950
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