Theorem esumeq2d 32005

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 2739	. 2 ⊢ (𝜑 → 𝐴 = 𝐴)
3		esumeq2d.1	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶)
4	3	r19.21bi 3134	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶)
5	1, 2, 4	esumeq12dvaf 31999	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

Syntax hints:   → wi 4   = wceq 1539  Ⅎwnf 1786  ∀wral 3064  Σ^*cesum 31995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-iota 6391  df-fv 6441  df-ov 7278  df-esum 31996

This theorem is referenced by:  esumeq2dv  32006  esumpad  32023  esumlef  32030  esumrnmpt2  32036  voliune  32197  omssubadd  32267  carsggect  32285  omsmeas  32290  dstrvprob  32438