Theorem esumeq12d 31901

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 480	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷)
4	1, 3	esumeq12dva 31900	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Σ^*cesum 31895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-iota 6376  df-fv 6426  df-ov 7258  df-esum 31896

This theorem is referenced by:  esumeq1  31902  esum2dlem  31960