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Theorem nllyeq 22838
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 2827 . . . . 5 (𝐴 = 𝐡 β†’ ((𝑗 β†Ύt 𝑒) ∈ 𝐴 ↔ (𝑗 β†Ύt 𝑒) ∈ 𝐡))
21rexbidv 3176 . . . 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 3213 . . 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 3418 . 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 22834 . 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 22834 . 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 2802 1 (𝐴 = 𝐡 β†’ 𝑛-Locally 𝐴 = 𝑛-Locally 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  βˆƒwrex 3074  {crab 3410   ∩ cin 3914  π’« cpw 4565  {csn 4591  β€˜cfv 6501  (class class class)co 7362   β†Ύt crest 17309  Topctop 22258  neicnei 22464  π‘›-Locally cnlly 22832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-nlly 22834
This theorem is referenced by: (None)
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