Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opnneilv Structured version   Visualization version   GIF version

Theorem opnneilv 48813
Description: The converse of opnneir 48811 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 48809), 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 3070 . 2 (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥(𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓))
2 opnneir.1 . . . . . . 7 (𝜑𝐽 ∈ Top)
3 neii2 23117 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑦𝐽 (𝑆𝑦𝑦𝑥))
42, 3sylan 580 . . . . . 6 ((𝜑𝑥 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑦𝐽 (𝑆𝑦𝑦𝑥))
54r19.41dv 48727 . . . . 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 3169 . . . 4 (𝜑 → (∃𝑦𝐽 ((𝑆𝑦𝑦𝑥) ∧ 𝜓) → ∃𝑦𝐽 (𝑆𝑦𝜒)))
136, 12syld 47 . . 3 (𝜑 → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦𝐽 (𝑆𝑦𝜒)))
1413exlimdv 1932 . 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 1778  wcel 2107  wrex 3069  wss 3950  cfv 6560  Topctop 22900  neicnei 23106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-top 22901  df-nei 23107
This theorem is referenced by:  opnneil  48814
  Copyright terms: Public domain W3C validator