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Theorem esumeq2sdv 34041
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 480 . 2 ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
32esumeq2dv 34040 1 (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  Σ*cesum 34029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-iota 6513  df-fv 6568  df-ov 7435  df-esum 34030
This theorem is referenced by:  ismeas  34201  isrnmeas  34202
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