Theorem esumeq12d 34030

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 34029	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Σ^*cesum 34024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-iota 6467  df-fv 6522  df-ov 7393  df-esum 34025

This theorem is referenced by:  esumeq1  34031  esum2dlem  34089