Theorem esumeq12d 31713

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

Hypotheses

Ref	Expression
esumeq12d.1	⊢ (𝜑 → 𝐴 = 𝐵)
esumeq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
esumeq12d	⊢ (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 esumeq12d

Step	Hyp	Ref	Expression
1		esumeq12d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		esumeq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
3	2	adantr 484	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷)
4	1, 3	esumeq12dva 31712	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷)

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2110  Σ^*cesum 31707

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 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-iota 6338  df-fv 6388  df-ov 7216  df-esum 31708

This theorem is referenced by:  esumeq1  31714  esum2dlem  31772