Theorem esumeq2d 34725

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 3258	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶)
5	1, 2, 4	esumeq12dvaf 34719	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  Ⅎwnf 1815  ∀wral 3080  Σ^*cesum 34715

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-iota 6501  df-fv 6555  df-ov 7433  df-esum 34716

This theorem is used by:  esumeq2dv  34726  esumpad  34743  esumlef  34750  esumrnmpt2  34756  voliune  34918  carsggect  35006  omsmeas  35011  dstrvprob  35160