Theorem esumeq12d 34368

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 485	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷)
4	1, 3	esumeq12dva 34367	1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Σ^*cesum 34362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-iota 6493  df-fv 6545  df-ov 7414  df-esum 34363

This theorem is referenced by:  esumeq1  34369  esum2dlem  34427