Theorem opnneilv 45641

Description: The converse of opnneir 45639 with different dummy variables. Note that the second hypothesis could be generalized by adding 𝑦 ∈ 𝐽 to the antecedent. See the proof for details. (Contributed by Zhi Wang, 31-Aug-2024.)

Hypotheses

Ref	Expression
opnneir.1	⊢ (𝜑 → 𝐽 ∈ Top)
opnneilv.2	⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))

Assertion

Ref	Expression
opnneilv	⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))

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

Step	Hyp	Ref	Expression
1		df-rex 3076	. 2 ⊢ (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥(𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓))
2		opnneir.1	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top)
3		neii2 21821	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥))
4	2, 3	sylan 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥))
5	4	r19.41dv 45626	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 ((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓))
6	5	expl 461	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 ((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓)))
7		anass 472	. . . . . 6 ⊢ (((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓) ↔ (𝑆 ⊆ 𝑦 ∧ (𝑦 ⊆ 𝑥 ∧ 𝜓)))
8		opnneilv.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))
9	8	expimpd 457	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ⊆ 𝑥 ∧ 𝜓) → 𝜒))
10	9	anim2d 614	. . . . . 6 ⊢ (𝜑 → ((𝑆 ⊆ 𝑦 ∧ (𝑦 ⊆ 𝑥 ∧ 𝜓)) → (𝑆 ⊆ 𝑦 ∧ 𝜒)))
11	7, 10	syl5bi 245	. . . . 5 ⊢ (𝜑 → (((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓) → (𝑆 ⊆ 𝑦 ∧ 𝜒)))
12	11	reximdv 3197	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐽 ((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
13	6, 12	syld 47	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
14	13	exlimdv 1934	. 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
15	1, 14	syl5bi 245	1 ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))

Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1781   ∈ wcel 2111  ∃wrex 3071   ⊆ wss 3860  ‘cfv 6340  Topctop 21606  neicnei 21810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-top 21607  df-nei 21811

This theorem is referenced by:  opnneil  45642