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Theorem nllyeq 23479
Description: Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllyeq (𝐴 = 𝐵 → 𝑛-Locally 𝐴 = 𝑛-Locally 𝐵)

Proof of Theorem nllyeq
Dummy variables 𝑗 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2830 . . . . 5 (𝐴 = 𝐵 → ((𝑗t 𝑢) ∈ 𝐴 ↔ (𝑗t 𝑢) ∈ 𝐵))
21rexbidv 3179 . . . 4 (𝐴 = 𝐵 → (∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴 ↔ ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐵))
322ralbidv 3221 . . 3 (𝐴 = 𝐵 → (∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴 ↔ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐵))
43rabbidv 3444 . 2 (𝐴 = 𝐵 → {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴} = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐵})
5 df-nlly 23475 . 2 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴}
6 df-nlly 23475 . 2 𝑛-Locally 𝐵 = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐵}
74, 5, 63eqtr4g 2802 1 (𝐴 = 𝐵 → 𝑛-Locally 𝐴 = 𝑛-Locally 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  cin 3950  𝒫 cpw 4600  {csn 4626  cfv 6561  (class class class)co 7431  t crest 17465  Topctop 22899  neicnei 23105  𝑛-Locally cnlly 23473
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-nlly 23475
This theorem is referenced by: (None)
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