Theorem esumeq1d 34491

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 2766	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐶)
4	1, 2, 3	esumeq12dvaf 34487	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2146  Σ^*cesum 34483

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-iota 6496  df-fv 6548  df-ov 7422  df-esum 34484

This theorem is used by:  esummono  34510  esumrnmpt2  34524  esumfzf  34525  hasheuni  34541  esum2dlem  34548  measvuni  34671  ddemeas  34693  omssubadd  34757  carsggect  34775