Theorem esumeq2dv 32002

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017.)

Hypothesis

Ref	Expression
esumeq2dv.1	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
esumeq2dv	⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

Distinct variable group:   𝜑,𝑘

Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)   𝐶(𝑘)

Proof of Theorem esumeq2dv

Step	Hyp	Ref	Expression
1		nfv 1921	. 2 ⊢ Ⅎ𝑘𝜑
2		esumeq2dv.1	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶)
3	2	ralrimiva 3110	. 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶)
4	1, 3	esumeq2d 32001	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1542   ∈ wcel 2110  Σ^*cesum 31991

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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-iota 6390  df-fv 6440  df-ov 7274  df-esum 31992

This theorem is referenced by:  esumeq2sdv  32003  esumle  32022  esummulc1  32045  esummulc2  32046  esumdivc  32047  esumsup  32053  measinb  32185  measres  32186  measdivcst  32188  measdivcstALTV  32189  cntmeas  32190  ddemeas  32200  omsval  32256  totprobd  32389