Theorem esumeq2d 34221

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 2740	. 2 ⊢ (𝜑 → 𝐴 = 𝐴)
3		esumeq2d.1	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶)
4	3	r19.21bi 3231	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶)
5	1, 2, 4	esumeq12dvaf 34215	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

Syntax hints:   → wi 4   = wceq 1547  Ⅎwnf 1790  ∀wral 3053  Σ^*cesum 34211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-iota 6441  df-fv 6493  df-ov 7359  df-esum 34212

This theorem is referenced by:  esumeq2dv  34222  esumpad  34239  esumlef  34246  esumrnmpt2  34252  voliune  34413  omssubadd  34484  carsggect  34502  omsmeas  34507  dstrvprob  34656