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Theorem opnneieqv 48509
Description: The equivalence between neighborhood and open neighborhood. See opnneieqvv 48510 for different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
Hypotheses
Ref Expression
opnneir.1 (𝜑𝐽 ∈ Top)
opnneilv.2 ((𝜑𝑦𝑥) → (𝜓𝜒))
opnneil.3 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
opnneieqv (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥𝐽 (𝑆𝑥𝜓)))
Distinct variable groups:   𝑥,𝐽,𝑦   𝑥,𝑆,𝑦   𝜒,𝑥   𝜑,𝑥,𝑦   𝜓,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem opnneieqv
StepHypRef Expression
1 opnneir.1 . . 3 (𝜑𝐽 ∈ Top)
2 opnneilv.2 . . 3 ((𝜑𝑦𝑥) → (𝜓𝜒))
3 opnneil.3 . . 3 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
41, 2, 3opnneil 48508 . 2 (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑥𝐽 (𝑆𝑥𝜓)))
51opnneir 48505 . 2 (𝜑 → (∃𝑥𝐽 (𝑆𝑥𝜓) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓))
64, 5impbid 212 1 (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥𝐽 (𝑆𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2103  wrex 3072  wss 3970  cfv 6572  Topctop 22913  neicnei 23119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-top 22914  df-nei 23120
This theorem is referenced by:  opnneieqvv  48510  sepnsepolem2  48521  sepnsepo  48522
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