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Theorem nllyeq 23388
Description: Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllyeq (𝐴 = 𝐡 β†’ 𝑛-Locally 𝐴 = 𝑛-Locally 𝐡)

Proof of Theorem nllyeq
Dummy variables 𝑗 𝑒 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2818 . . . . 5 (𝐴 = 𝐡 β†’ ((𝑗 β†Ύt 𝑒) ∈ 𝐴 ↔ (𝑗 β†Ύt 𝑒) ∈ 𝐡))
21rexbidv 3175 . . . 4 (𝐴 = 𝐡 β†’ (βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐴 ↔ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐡))
322ralbidv 3215 . . 3 (𝐴 = 𝐡 β†’ (βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐴 ↔ βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐡))
43rabbidv 3437 . 2 (𝐴 = 𝐡 β†’ {𝑗 ∈ Top ∣ βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐴} = {𝑗 ∈ Top ∣ βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐡})
5 df-nlly 23384 . 2 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐴}
6 df-nlly 23384 . 2 𝑛-Locally 𝐡 = {𝑗 ∈ Top ∣ βˆ€π‘₯ ∈ 𝑗 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π‘—)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝑗 β†Ύt 𝑒) ∈ 𝐡}
74, 5, 63eqtr4g 2793 1 (𝐴 = 𝐡 β†’ 𝑛-Locally 𝐴 = 𝑛-Locally 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058  βˆƒwrex 3067  {crab 3429   ∩ cin 3946  π’« cpw 4603  {csn 4629  β€˜cfv 6548  (class class class)co 7420   β†Ύt crest 17402  Topctop 22808  neicnei 23014  π‘›-Locally cnlly 23382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-nlly 23384
This theorem is referenced by: (None)
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