Theorem esumeq2d 34038

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 2738	. 2 ⊢ (𝜑 → 𝐴 = 𝐴)
3		esumeq2d.1	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶)
4	3	r19.21bi 3251	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶)
5	1, 2, 4	esumeq12dvaf 34032	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

Syntax hints:   → wi 4   = wceq 1540  Ⅎwnf 1783  ∀wral 3061  Σ^*cesum 34028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-iota 6514  df-fv 6569  df-ov 7434  df-esum 34029

This theorem is referenced by:  esumeq2dv  34039  esumpad  34056  esumlef  34063  esumrnmpt2  34069  voliune  34230  omssubadd  34302  carsggect  34320  omsmeas  34325  dstrvprob  34474