Theorem esumeq12d 33999

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 33998	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  Σ^*cesum 33993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-iota 6527  df-fv 6583  df-ov 7453  df-esum 33994

This theorem is referenced by:  esumeq1  34000  esum2dlem  34058