Theorem esumeq1d 32242

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 2737	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐶)
4	1, 2, 3	esumeq12dvaf 32238	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540  Ⅎwnf 1784   ∈ wcel 2105  Σ^*cesum 32234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-opab 5152  df-mpt 5173  df-iota 6425  df-fv 6481  df-ov 7332  df-esum 32235

This theorem is referenced by:  esummono  32261  esumrnmpt2  32275  esumfzf  32276  hasheuni  32292  esum2dlem  32299  measvuni  32421  ddemeas  32443  omssubadd  32508  carsggect  32526