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Theorem esumeq2d 30406
Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016.)
Hypotheses
Ref Expression
esumeq2d.0 𝑘𝜑
esumeq2d.1 (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)
Assertion
Ref Expression
esumeq2d (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)

Proof of Theorem esumeq2d
StepHypRef Expression
1 esumeq2d.0 . 2 𝑘𝜑
2 eqidd 2759 . 2 (𝜑𝐴 = 𝐴)
3 esumeq2d.1 . . 3 (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)
43r19.21bi 3068 . 2 ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
51, 2, 4esumeq12dvaf 30400 1 (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wnf 1855  wral 3048  Σ*cesum 30396
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 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-mpt 4880  df-iota 6010  df-fv 6055  df-ov 6814  df-esum 30397
This theorem is referenced by:  esumeq2dv  30407  esumpad  30424  esumlef  30431  esumrnmpt2  30437  voliune  30599  omssubadd  30669  carsggect  30687  omsmeas  30692  dstrvprob  30840
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