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Theorem esumeq2dv 30434
Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017.)
Hypothesis
Ref Expression
esumeq2dv.1 ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
esumeq2dv (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Distinct variable group:   𝜑,𝑘
Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)   𝐶(𝑘)

Proof of Theorem esumeq2dv
StepHypRef Expression
1 nfv 1994 . 2 𝑘𝜑
2 esumeq2dv.1 . . 3 ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
32ralrimiva 3114 . 2 (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)
41, 3esumeq2d 30433 1 (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  Σ*cesum 30423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-iota 5994  df-fv 6039  df-ov 6795  df-esum 30424
This theorem is referenced by:  esumeq2sdv  30435  esumle  30454  esummulc1  30477  esummulc2  30478  esumdivc  30479  esumsup  30485  measinb  30618  measres  30619  measdivcstOLD  30621  measdivcst  30622  cntmeas  30623  ddemeas  30633  omsval  30689  totprobd  30822
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