Theorem esumeq1d 34332

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 2763	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐶)
4	1, 2, 3	esumeq12dvaf 34328	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560  Ⅎwnf 1803   ∈ wcel 2142  Σ^*cesum 34324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-iota 6477  df-fv 6529  df-ov 7399  df-esum 34325

This theorem is referenced by:  esummono  34351  esumrnmpt2  34365  esumfzf  34366  hasheuni  34382  esum2dlem  34389  measvuni  34511  ddemeas  34533  omssubadd  34597  carsggect  34615