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Theorem esumeq12d 30936
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 473 . 2 ((𝜑𝑘𝐴) → 𝐶 = 𝐷)
41, 3esumeq12dva 30935 1 (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  Σ*cesum 30930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-mpt 5010  df-iota 6154  df-fv 6198  df-ov 6981  df-esum 30931
This theorem is referenced by:  esumeq1  30937  esum2dlem  30995
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