Theorem esumeq2d 34427

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016.)

Hypotheses

Ref	Expression
esumeq2d.0	⊢ Ⅎ𝑘𝜑
esumeq2d.1	⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶)

Assertion

Ref	Expression
esumeq2d	⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

Proof of Theorem esumeq2d

Step	Hyp	Ref	Expression
1		esumeq2d.0	. 2 ⊢ Ⅎ𝑘𝜑
2		eqidd 2764	. 2 ⊢ (𝜑 → 𝐴 = 𝐴)
3		esumeq2d.1	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶)
4	3	r19.21bi 3257	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶)
5	1, 2, 4	esumeq12dvaf 34421	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

Syntax hints:   → wi 4   = wceq 1570  Ⅎwnf 1813  ∀wral 3079  Σ^*cesum 34417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-iota 6492  df-fv 6544  df-ov 7413  df-esum 34418

This theorem is referenced by:  esumeq2dv  34428  esumpad  34445  esumlef  34452  esumrnmpt2  34458  voliune  34619  omssubadd  34690  carsggect  34708  omsmeas  34713  dstrvprob  34862