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Theorem opnneilv 48705
Description: The converse of opnneir 48703 with different dummy variables. Note that the second hypothesis could be generalized by adding 𝑦𝐽 to the antecedent. See the proof for details. Although 𝐽 ∈ Top might be redundant here (see neircl 48701), it is listed for explicitness. (Contributed by Zhi Wang, 31-Aug-2024.)
Hypotheses
Ref Expression
opnneir.1 (𝜑𝐽 ∈ Top)
opnneilv.2 ((𝜑𝑦𝑥) → (𝜓𝜒))
Assertion
Ref Expression
opnneilv (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦𝐽 (𝑆𝑦𝜒)))
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 opnneilv
StepHypRef Expression
1 df-rex 3069 . 2 (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥(𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓))
2 opnneir.1 . . . . . . 7 (𝜑𝐽 ∈ Top)
3 neii2 23132 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑦𝐽 (𝑆𝑦𝑦𝑥))
42, 3sylan 580 . . . . . 6 ((𝜑𝑥 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑦𝐽 (𝑆𝑦𝑦𝑥))
54r19.41dv 48651 . . . . 5 (((𝜑𝑥 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝜓) → ∃𝑦𝐽 ((𝑆𝑦𝑦𝑥) ∧ 𝜓))
65expl 457 . . . 4 (𝜑 → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦𝐽 ((𝑆𝑦𝑦𝑥) ∧ 𝜓)))
7 anass 468 . . . . . 6 (((𝑆𝑦𝑦𝑥) ∧ 𝜓) ↔ (𝑆𝑦 ∧ (𝑦𝑥𝜓)))
8 opnneilv.2 . . . . . . . 8 ((𝜑𝑦𝑥) → (𝜓𝜒))
98expimpd 453 . . . . . . 7 (𝜑 → ((𝑦𝑥𝜓) → 𝜒))
109anim2d 612 . . . . . 6 (𝜑 → ((𝑆𝑦 ∧ (𝑦𝑥𝜓)) → (𝑆𝑦𝜒)))
117, 10biimtrid 242 . . . . 5 (𝜑 → (((𝑆𝑦𝑦𝑥) ∧ 𝜓) → (𝑆𝑦𝜒)))
1211reximdv 3168 . . . 4 (𝜑 → (∃𝑦𝐽 ((𝑆𝑦𝑦𝑥) ∧ 𝜓) → ∃𝑦𝐽 (𝑆𝑦𝜒)))
136, 12syld 47 . . 3 (𝜑 → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦𝐽 (𝑆𝑦𝜒)))
1413exlimdv 1931 . 2 (𝜑 → (∃𝑥(𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦𝐽 (𝑆𝑦𝜒)))
151, 14biimtrid 242 1 (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦𝐽 (𝑆𝑦𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1776  wcel 2106  wrex 3068  wss 3963  cfv 6563  Topctop 22915  neicnei 23121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-top 22916  df-nei 23122
This theorem is referenced by:  opnneil  48706
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