Theorem esumeq1d 31404

Description: Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)

Hypotheses

Ref	Expression
esumeq1d.0	⊢ Ⅎ𝑘𝜑
esumeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
esumeq1d	⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶)

Proof of Theorem esumeq1d

Step	Hyp	Ref	Expression
1		esumeq1d.0	. 2 ⊢ Ⅎ𝑘𝜑
2		esumeq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		eqidd 2799	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐶)
4	1, 2, 3	esumeq12dvaf 31400	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  Σ^*cesum 31396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-iota 6283  df-fv 6332  df-ov 7138  df-esum 31397

This theorem is referenced by:  esummono  31423  esumrnmpt2  31437  esumfzf  31438  hasheuni  31454  esum2dlem  31461  measvuni  31583  ddemeas  31605  omssubadd  31668  carsggect  31686