Theorem esumeq12dva 34228

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.) (Revised by Thierry Arnoux, 29-Jun-2017.)

Hypotheses

Ref	Expression
esumeq12dva.1	⊢ (𝜑 → 𝐴 = 𝐵)
esumeq12dva.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷)

Assertion

Ref	Expression
esumeq12dva	⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷)

Distinct variable group:   𝜑,𝑘

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

Proof of Theorem esumeq12dva

Step	Hyp	Ref	Expression
1		nfv 1922	. 2 ⊢ Ⅎ𝑘𝜑
2		esumeq12dva.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		esumeq12dva.2	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷)
4	1, 2, 3	esumeq12dvaf 34227	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  Σ^*cesum 34223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-iota 6445  df-fv 6497  df-ov 7363  df-esum 34224

This theorem is referenced by:  esumeq12d  34229